Currently the way things work is that when a pet's done a training session, you get taken to a new page to be told how many points your pet goes up, then have to click the back link, then have to scroll down to find your pet again to put it back into training, which takes to to a new page again, and then have to click the back link again to get back into the training center to train another pet.
That's...a lot of clicking.
Instead, why not have pop-up boxes? When you go to take your pet out of training once it's done, the page stays right where it is. A pop-up box comes up telling you what your pet earned. Then, it allows you to put your pet back into training, allowing you to choose what you want to train. Once done, you just close the pop-up.
That's how it'd work in theory. I don't know if it's doable, but it's a thought. If anyone has anything better to offer, feel free.

Personally I wouldn't like this. I train all my 20 pets, and this way it'd take me too much time to put them on a new training session and getting a pup up for every pet. I can see how it can be really useful for people who train 1 or 2 pets, but really annoying for people who like to train all of their pets :(
I think the current system is fine as is. Popups drive me to drink, so the less the better.
You'd have fewer page loads, but I don't see how this would reduce the amount of clicking necessary. It sounds like the same amount of work either way.

I'm not seeing how this makes things more fun or easier. Are you thinking there'd be a default you could just press Return for in the popup?
Pageloading time is usually not an issue for me, so dealing with a popup sounds like the same amount of work for less REs. I train all fifteen of my pets, and this gives me a decent chance to nab something random as a bonus, which wouldn't happen if it were popup.