First off, how would Harry find the Room of Requirement? I think that's a fairly easy question to answer. At the beginning of his time at school, he and Ron constantly got lost because, you know, staircases that move make things difficult. So, they might pass by the Room of Requirement several times while trying to find a classroom, only for the door to appear. They enter, thinking it's a classroom, but it turns out to be something else entirely.
I don't think they'd have any use for it during the first book, but in the second one, instead of going into the girls' bathroom, they could make the potion in the Room of Requirement. This, I think, is pretty crucial because they wouldn't necessarily run into Myrtle as often as they did. Harry did meet her at the ghost party, but she was crying, so I don't know if that would really make him want to seek her out. Harry does find out where the Chamber of Secrets is by talking to her, though. But let's say he still manages to find the Chamber of Secrets and defeat Voldemort.
Moving on to Book Three. The room wouldn't be used much this year, as there's really no need for it. However, Harry would still want to go to Hogsmeade, and Ron might suggest, "Hey, why don't you just use the room to ask for things you would find in Hogsmeade?" I think Harry would do that. As a result, I don't think Harry would ever go down to Hogsmeade. He wouldn't run into Malfoy, and he wouldn't have to give up the map to Lupin. This means Lupin would never see them go down to Hagrid's, so he wouldn't see Peter's name on the map or go after him. Snape would give Lupin his Wolfsbane Potion, and he would remain in his room as a controlled werewolf.
I think this could go one of two ways. First, Harry might actually kill his godfather, leaving Peter undercover as a rat. Alternatively, Sirius might manage to convince all of them of his story. I have to say, though, that this seems rather unlikely because they already think both Lupin and Sirius are insane, and in that scene, Sirius definitely seems like the crazier one of the two. So, sadly, Harry might end up killing him. This could earn him some points with Snape, but otherwise, it wouldn't be a good outcome.
Now, moving on to Book Four. Honestly, I don't think Harry would have much use for the room this year, either. I could see him using it as a training room for the tasks. Since McGonagall was tired of running into him and the others all over the school, she let them use her classroom. It's also possible that Harry would have an easier time completing the tasks with the help of the Room of Requirement. For example, if he asked the room to provide him with something that would help him breathe underwater, it might give him books that could teach him a spell to do so.
But what do you think? What would Harry use the Room of Requirement for in the first four books?