What parts of the body should be trained? How to be ready for a hard push?
So, there's a few answers, the correct part depends on what you're really asking.
I don't want to be knocked back
Well, there's a few arts that end up practicing "rooting" and folks can bounce you off if you try to push them. That's going to be about 10-15 years of training your legs, spine and mostly how to efficiently use it against someone else's center of balance on near instant contact. This is going to be stuff like a really good Tai Chi or Bagua person.
It's a neat skill, though I'm not sure it's be something worth the time/effort for what you're asking.
You can of course, try doing stuff like Sumo, or American Football training which focuses a lot on forward push/resist stuff, but in both of those cases the pushing is much more formalized and clear when/where it will happen, which isn't like how it happens in a fight.
I don't want to be knocked down
The best way to not be knocked down is to avoid the push, or, if it makes contact, do a small hop backwards to keep your feet under you while you're being pushed. Naturally, this means you "recoil" or are pushed back. You can also rotate and take off the force if it's not direct to your center.
I don't want to be hurt when pushed
All the stuff for not being knocked down, as above, plus falling training. Best thing for me was Judo training. I spent about 2 weeks learning nothing but falling techniques, and every class began with falling training for everyone as a warm up. I haven't taken Systema, but they look like they also have some pretty good falling skills as well.
Assuming that you're talking about being pushed in the manner of posturing males ("OH YEAH, BRING IT?!?!", etc), then you have a couple options:
Now, with all that said: I understand that you, or anyone else, may disagree with some of these statements. I'd be happy to hash them out (probably in the chat) with anyone who has questions.
Alternatively, find a patient friend (or a patient enemy, I suppose) and have them push you, over and over and over, until you've figured out for yourself what your response should be. How do you think martial arts started in the first place? :D
Being pushed is typically a kind of posturing - an assertion of strength and dominance, rather than an actual attack. So, learning to deal with it well can be useful at preventing further escalation in a bullying situation - undermining the dominance the other person expects to feel - but isn't very useful in an actual fight.
Whatever you want to do, you need to practice at least a few dozen times with a friend.
I assume you're thinking of a double handed chest to the chest, or a single handed push to the shoulder, chest or even face. A push is a directional delivery of power over time, with a range of 30-50cm or so, and the aggressor needing to be reasonably close to begin the movement.
There are two aspects to this, assuming that by "avoiding recoil" you mean you don't want to move backwards.
To reducing their effectiveness, you need to rotate away from the place they apply force without a stiffness in your posture that communicates their force back to your centre of mass. So, if they go to push you on one shoulder and you stiffen up and seek to stay front facing, you'll tend to be pushed back. If you relax the shoulder they shove and let the other shoulder swing forwards, then they'll have little 'net effect on your centre of mass. If they target your centre better, then you should look to turn either shoulder forwards, letting their hand(s) slip across your chest rather than through it. If they're shoving with their right hand, you want your left shoulder coming forwards / right shoulder backwards. You can use your hands on the outsides of their forearms to help divert their push past you.
You can also effectively reduce the range over which they can develop power by leaning in slightly while keeping your hips back... that way they have to push your upper body back further before it becomes difficult to keep your hips/centre of mass stationary. So, it may look like the shove moves you back effectively, but you are less likely to lose your balance afterwards or keep stumbling backwards. Be careful of being pulled forwards though, or put into a headlock.
Further, if you stay in close to your opponent, they'll have trouble generating as hard an initial impact... they may manage a strong push, but will have a little more trouble having as much of a bruising "hit" aspect to it. They'll have to make contact earlier - before they've generated as much forward momentum - and you have a chance to be "sensitive" to their direction and timing of force, making it easier to twist away from the thrust, or even suddenly jerk back into them at an unexpectedly close distance before they've got their push focused.
Another option is to rest your arms atop of their elbows, so it's harder for them to extend through the push. That's most practical if they keep their elbows low, while a defection works well even with elbows high. You can do either of these things without seeming overly aggressive or necessarily escalating the violence... they show a degree of control which psychologically may persuade the attacker that escalation might be riskier than they thought.
In terms of actively meeting force with resistance, you want to either start of come to rest with one foot forward of the other - preferably so their stronger/dominant hand will slide across your chest rather than push through it. Keep the front foot facing forwards, and the back leg's knee turned more forwards too, and bent. Imagine you were about to push a truck with a flat battery along the road... you'd probably use a similar stance. It's nice not to necessary start with too strong a stance though... you want to avoid some of the force, allow other force to rotate your body so one shoulder goes back and the other forward, and only directly oppose whatever's left or find a strong stance to settle back in to to stop any momentum they've impacted - ideally when you're already at the edge of or out of their range of movement for the push.
If things do get - or are obviously about to get - seriously nasty, just abandon all this... it's just delaying the inevitable. If they're about to shove you hard, step back and pull them forwards, use a joint lock if you know how, jab them in the throat before they shove, palm to nose, elbow up under the chin, knee or front kick to the groin, kick hard to the kneecap etc..
So ultimately, there's a lot you can do with timing and technique - I'd worry less about building specific muscles for this than just getting used to moving well in the situation, again practicing with a partner.
If you're asking about strength work then the answer is, as always, get strong first. Don't try to tailor your strength training to a specific task if you're weak, because that approach will keep you weak. Do whole-body general strength training, like pull-ups and heavy deadlifts.
The rest of resisting or avoiding a bad response to a push is technique training, or more generally getting used to someone disrupting your balance. For that, you should be sparring hard in some sort of grappling ruleset where people are going to push you. Wrestling, judo, san da, sumo, and SAMBO stand out as excellent options, as they involve hard sparring with throws and shoves. BJJ would be an OK but lesser option due to its lack of focus on the stand-up portion of the fight. If you're really keen on training specifically for being pushed in a self-defense scenario (which is a bit silly, as it's overspecialization), then judo or sumo might be best since the posture used in those arts are the most upright and therefore the most similar to how one would be standing when pushed during an ape dominance interaction.
People who spar hard in grappling arts don't usually need to ask these kinds of questions, because dealing with being pushed is second nature after weekly rough-and-tumble stand-up grappling. (This occurs also with techniques for dealing with wrist grabs. Wrestlers just deal with it.)
Momentum and collisions are what you are looking at. Collisions can be either elastic or inelastic. The former is when you recoil if pushes, the latter is when your body absorbs most the energy of the push.
Now, what you are asking is how to effectively absorb the energy. Well, momentum is conserved in a collision, so you have to provide equal momentum in the opposite direction to the push. Alternatively, you can absorb the energy which will likely translate as damage to your body. You can do that by standing your ground. Clearly, the former should be preferred.
How do you achieve equal momentum in the opposite direction to the push? Well, you have mass and you have velocity. The latter is the one you want to improve on due to kinetic energy. Which means you have to move towards the push. By doing so you are increasing the total energy of the system. This is bad because some of that energy will translate into damage to both bodies: bruises, bones fractures, concussion, etcetera. After all, our bodies are not rigid ones. Note that damage can be due to the initial collision or the collision with whatever you collide with behind you: wall, floor, etcetera.
So, neither options is really avoiding damage to yourself. Maybe either avoiding or blending with the push are better ways to deal with a push than trying to resist it.
In addition, offensive means imparting energy (kick, punch, throw, ...) into the target and defensive means minimising the energy (absorb, block, deflect, avoidance, ...) you receive as part of the attack. It is all about physics.
It's kind of hard to tell what you're asking for here. Are you asking for advice on what ways you can deal with a push after you've been pushed and you're just sort of ejected away from someone while trying to regain your balance and stop? Or are you asking for advice on what to do at the moment someone begins to push? Or are you asking for advice on what to do once you realize a push is or might be coming?
All of those are legitimate questions. Pushing is one of the most common things you'll encounter if you're in a bad situation. It's often the prelude to more violent actions. How you respond during this phase of the fight is critical.
Prior to being pushed, you should realize the situation is getting tense, and your body language and words should communicate that you don't want to fight. That can stop the fight from happening right there. As you're doing this, you're keeping some distance between you and whoever is threatening you. So back away. Keep your hands up with your palms open towards the guy, just like you see in cop shows where the cops ask people to raise their arms. This is calming and is also the universal sign that you don't want to fight. But it's also great for you, because if the guy comes at you, you've got your arms in a good position to quickly defend. Also, turn your body sideways slightly, so you're not facing the guy squarely. This is so that if he lunges directly into you, you can pivot quickly and take yourself out of his path. It also presents less targets for him to hit or grab onto.
So there's plenty of things you can do before even knowing what's going to happen.
Next, if the guy attacks you with a sudden rush forward, you have some time to get off the line and dodge like I said above. Your arms should quickly move downwards to protect your body and meet his force while not over-extending. At the same time, you're hunching your body forward, getting the legs spread out into a forward wrestling stance, and preparing to sprawl and/or shoot in for a take-down.
If he manages to get both hands on your chest and begins to push while you're not prepared for it (just standing up straight), you are in trouble. In this stance, you are very easy to push. You'll go flying backwards. Recovery is critical, because you're either going to trip and fall on your back, or you're going to spend a lot of time flailing around trying to get your balance. Either way, this means you're vulnerable to anything your attacker wants to do to you for a short period of time while you're trying to recover. So this situation is very hard to turn around to your advantage. Your strategy here should be to recover as soon as possible and get back to a strong position.
Some martial arts I've found fairly good at helping people deal with pushing are: Tai Chi, Brazilian Jiujitsu, Judo, and Wrestling. I would think Sambo and other grappling styles would also be very good, but I have no direct experience with them. Of those, I'm giving the edge to Brazilian Jiujitsu, because it has a full game plan from the very beginning. But Tai Chi also has some good material for dealing with pushes and could very easily complement any other of the styles I mentioned. I wouldn't take Tai Chi alone, however, because I just don't see it as having a complete overall strategy, especially if things go wrong.
As Dave suggests, get stronger by doing whole body weight lifting. Strength training will be helpful. Especially look into the "weight sled" apparatus. And maybe bungee cord harnesses. Those train your forward "push" strength, especially in the legs, but also in the back, abdomen, and trapezius. You'll also need to work on your "pull" exercise to balance yourself out so you avoid muscle imbalances later on.
It's all about your core strength.use your core,then decide whether it's better to deflect with a twist of the torso or absorb ,with one step back. But the point is, no strategy will be effective without core strength. There is a huge amount of information on the inter web on core training. You just have to do it …On a daily basis.