Short answer: You are approaching your problem the wrong way. Don't try to memorize the combos, try to understand them. Deconstruct the combos. Make the combo personal - ask "Why?" for every strike. And then, when you know the "Why?", drill the combos again and again.
I can never remember when they're said in class.
Well, first of all, you should know that (almost) nobody can remember long combos the first few times they do it. We start to get the hang of it after may be the 3rd time.
The problem is, many students try these combos in the class and forget about them once they step outside the gym. If you are serious about your art, note them down after the class and look at them before going to bed. That is what I do. I try to visualize every combo in my head when I am in the bus, having a coffee, walking, or before going to bed. If you feel writing it down is pretentious, turn on your recording app on your phone, and go "Combo 1 Jab, Slip left, Shovel hook to the body, Overhand right, ...". Later, just put your headphones on and start visualizing.
I know the "basics" (I think) like jab cross hook. I understand it goes left right left right most of the time and body head body head ... I don't know if it's just a memory problem or something else but I seem to be the worst at this.
You don't say if this is a competitive MT/KB class or one of those cardio KB classes. If its a cardio KB class, then just keep doing what you are doing and with time you will be able to get the combos down.
If you are doing competitive MT/KB - meaning that you want to improve in the art, you truly love it, and you want to keep doing it for as long as you can - then please take a moment and read this.
I understand it goes left right left right most of the time and body head body head.
No, that is wrong (of course, in my humble opinion). Why are you doing combos? What is the purpose of a particular combo? Before you do a combo, you need to know what each strike does. You have to genuinely understand what you are doing and what each movement accomplishes.
Many people don't understand this. You can have a great fighter, a champion, as your coach, but if they don't explain to you why you are doing a particular thing (e.g., a combo, a movement, a drill, etc.), then there is no use for that combo, and you are likely to never succeed.
A good teacher will always make the effort to explain why you are doing something, and won't just tell you to memorize a long combo. Often, in a class full of 40 students, coaches will not explain things in detail, because not many are really interested in knowing the art by heart. Most students are there to just get a good cardio session in, but it's these guys who make up the bulk of the fees. I digress.
Sometimes it is up to you to find out why you are doing a particular combo. Study the art, ask your coach, and look at Youtube videos. When you truly understand the basics of small combos, the long combos in the class will start to make sense. Even before you do it the first time, you will know - why, for example, the Overhand right follows the Left Hook to the body. When you learn to put a reason behind a strike, everything will become simpler.
Now, let's talk about some combos and try and understand the reason behind each strike. Assume both you and your opponent, Arnold, are orthodox.
Combo 1: Jab, Cross (Straight right), Left Hook to the head, Right Leg Kick (using your back leg to his front leg).
What is the goal of this combo? To me, and to many, it's the finishing Leg kick (chopping down the tree!). What is the purpose of the Jab + Cross? What is the purpose of the Left Hook to the head?
The Jab + Cross gets Arnold to parry or cover up to protect their face. Parrying the straight punches or covering up makes Arnold vulnerable/open to a powerful Left Hook to the head.
Whether the Left Hook to the head lands flush or gets blocked (Arnold brings his right hand up on the side) is not important. If it lands flush, excellent. If not, then the Left Hook serves a very important purpose.
The Left Hook to the head dramatically shifts Arnold's body weight on his left side - on his left leg. When Arnold's weight is on his left leg, that is when your Right Leg Kick is the most lethal and does the most damage. Therefore, its not just a mere left right left combo that you memorize, but its a very important foundation of many other combos that you need to understand.
Combo 2: Jab, Cross, Left Hook to the head, Cross + Switch stance to southpaw (don't skip but slide your back leg past your front), Left Body Kick - Double up (using your Left leg which is now the back leg).
After the first combo (Combo 1), an average Joe will think that when you do the Jab + Cross + Left Hook to the head, you are going to throw in the Right Leg Kick.
But this time, you are going to follow up the Left Hook to the head with the Cross, going straight through Arnold's high guard. The goal here is to land at least one Left Body Kick to the liver.
The Left Hook to the head followed by the Cross will automatically cause Arnold to put more tension on his guard and get him to think about the "head hunting". He will leave his liver open for your devastating Left Body kicks.
Your second Cross also serves another purpose. It hides your stance switch and allows you to close the distance to land your shin to Arnold's liver area. Why then switch the stance? Why not just throw a Right Body Kick?
Well, the Left Body Kick is more effective if you can land it in the liver area. It can completely shut your opponent down if landed correctly. The doubling up will have to be fast. If landed correctly, the first kick will shut their body down (freezing them), the second should knock them down. Of course, that is the basic theory.
Your example Combo: Jab, Cross, Jab, Switch kick, then Body Hook, Head Hook right
I don't understand your combo fully. But let's give it a try.
The Jab + Cross gets their guard up high. The second Jab hides the stance switch and keeps pressure on his right hand so he keeps that side of the liver open.
The Left Body Kick (switch kick) hits Arnold in the liver. You follow this up with an enormous Left Hook to the body. Wait, why the switch kick and the Left Hook to the body you ask?
When you throw the Left Body Kick, your Left leg lands in front in this case (you are back to your orthodox position). If you simply bend your body to your left a little while bringing down your Left leg (after the kick) with that momentum, you are now loaded up for a devastating Left Hook to the body.
This Left Hook to the body is so painful that Arnold brings his hands down to cover his body. This makes an opening for the goal: the Overhand right (or the Right Hook to the head). This last strike is often hard to see coming because your opponent is too caught up with the major power shots going on in his left side.
While this might be overwhelming to understand in the beginning, it will make a lot of difference in your training. Think of a combo this way: Jab + Cross occupies his hands, Left Hook to the Head gets him to shift his weight on his left leg. Left Hook to the body brings his hands down leaving his head open for a Left Hook to the head, any Right Hand strikes, or a Right Head Kick.
Watch lots of fight breakdown videos on Youtube. Once you start visualizing how your opponent reacts to your strikes, long combos will start making sense. You won't really struggle that bad with them in the class. You will be able to distinguish between good, effective combos and really bad ones that serve no purpose.