My instructor has been making us practice the forward roll, and mentioned that once we are confident with it, we would move on to learn the back roll.
Researching the subject on the Internet, I find that most ukemi basics videos start drilling the back roll, and only after introduce the front roll as just a reverse of the back roll.
Which one should we learn first?
What you describe is the usual practice: Backwards first, then sidewards (both without rolling), then backwards rolling, and forwards as the last one.
As a German Judo coach, I tried to find some English guidelines that define the order of teaching ukemi, but could not find one.
In the German guidelines for teaching Judo, it says on page 16:
For learning O-goshi or Uki-goshi on the training level of 8th kyu breakfalls to the front are not compulsory. More important at that point are learning the correct landing position on the side and proper use of the slapping arm. (translation mine)
Accordingly, the mandatory ukemi for the first belt are only sidewards and backwards (cf. p. 20).
Rolling movements are hard for many beginners. I teach basic levels (for those who are afraid) always sitting on the ground and without rolling, see also my post here for further aspects and reasons.
As of rolling forwards vs. rolling backwards: They have a fundamental difference - the rolling of the backwards ukemi evolves naturally and can be achieved by gradually putting more momentum into the movement and making the back rounder.
This means each student can extend their own comfort zone gradually in their own time. The forward movement works fundamentally different. You may start at the ground, but apart from that, there is no "gradual" approach. You either manage your guts and go upside down or not. Without lifting of the hips, this ends up all messy.
In other words: Even with the nicest of all approaches, the forward rolling ukemi is by far the most complex and hardest to learn of the standard (i.e. beginners') ukemi.
Starting with it can work in a group with basic coordination and gymnastics abilities (i.e. I can assume everyone is able to do a mediocre somersault). It can be a great motivator to start with this and when they managed it, point out that this was the hardest one. But if I suspect that I have group members that will have a hard time learning it, I'd always start with backwards and sidewards - no rolling at all.
You should learn whichever one your instructor tells you to learn first. It makes no difference, they're both basic techniques that you'll eventually need to learn.
I will say that for beginners, meeting the mat can be a nerve-wracking event, and a forward roll gives the student a chance to see the mat as they are falling and rolling. A back roll requires a little bit of a leap of faith, as it were. So I think forward rolls are easier for beginners to learn.
But if I saw a student being taught a back roll as their first ukemi technique, I wouldn't have a problem with it.
What's most commonly taught first varies by martial art: in Aikido (and Hapkido for whatever that's worth / appreciate you're using Japanese terminology) training, it's very, very common to roll forwards out of various joint locks so that's a day one technique, while as a beginner you're more likely to be taught a back break-fall than an actual roll. I defer to Philip's insights into common judo practice and the order of introduction there. The question doesn't mention any particular art.
Even within an art, there may be variation for all sorts of reasons: e.g. if the other beginners learnt one technique the week before you started, you might find yourself learning the other for a bit before they cycle back. One instructor might do it differently to another.
All up, it's such a trivial matter that it's not worth second-guessing your instructor. That (s)he's there to guide your form is what matters. In a year, which you learnt first won't have made any difference.
Adding to Phillip's commentary, my experience in Aikido and Jujitsu, as well as a handful of arts where grappling isn't primary, breakfalls are generally taught first because they have fewer moving parts. After that, backwards (over one-shoulder) rolls are generally taught because it's intuitive (it starts with a very natural movement of relaxing your knees and dropping to a crouch before rolling backwards) and starts with less vulnerable parts of your body hitting the mat (forward rolls, even if started low and already threading your arm through your legs, still involve a degree of throwing yourself headfirst at the mat).
This is the opposite of what I've experienced in tumbling training, but I think that has to do with a difference in technique. Rolls in grappling tend to be diagonal, shoulder to opposite hip. Tumbling rolls are square, over your spine, and it's easier to hold your body weight up with your arms in the forward roll at first then it is for most people to take that leap of faith to roll backwards over your neck as you learn to plant your hands and push hard enough to lift yourself off of the mat to have your head clear.
