Within the context of Classic, the method most frequently proposed is to re-tune bosses to make them harder. Rather than copying the numbers, re-create the relative difficulty. Kind of like, when you first start working out, you use small weights. When you get stronger, you lift bigger weights. The WoW community is more knowledgeable, more experienced, has better hardware, etc. We're "stronger." So give us tougher bosses. There are a lot of forms that could take. Bosses were constantly re-tuned during vanilla anyway, so if Ragnaros does 10% more damage or something, is that really a problem? I don't think it is. What about resistance gear? back in the day, collecting resist gear was a thing people did. But the numbers and strategies are so well known today that only the tank really needs it anymore. Bosses could be re-tuned with more resistible elemental damage so that's important for entire raids to gear appropriately. That would be entirely keeping with the spirit of vanilla even if the actually numbers sitting in a database somewhere were different than they were ~13 years ago. If you want to go a little farther, a few people have proposed adding new abilities to old bosses, adjusting their timing, etc.
Each battle that you win will yield experience for each pet that participated. However, deceased companions will not receive any experience (so make an effort to keep them alive by swaping in a different pet). Experience gains are based on your pet's level compared to the wild pet. Defeating a higher level opponent will grant more experience than a much lower level one, but be ready for a tough fight!