I can't make it go faster or easier, but here's a few things you can do things to make it feel that way.
- Set a realistic goal. I think mine was 10 wins a day, and after that, I stopped, even if I was on a winning streak. It kept the task from feeling too overwhelming.
- Run more than 1 team. I mainly used a darkness team but when I got tired of it, or just ran into a string of losses, I swapped to my secondary team (it was a swap/minefield team).
- Do something else while pet battling, if you can multitask. I like watching videos in the background, or just reading comics, especially if the other player is slow with their turns. You might end up making a silly mistake here and there due to distraction, but it does make the time pass by a lot faster.
- This last one probably won't work for everyone (and raiding is pretty much over for this expansion) but my favorite time to queue for pet battles was while sitting out for someone in raid, since you can queue while in a raid group. So since I had to stay online anyhow, it was a good way to fill up that time. I also did regular tamer battles during those breaks as well.