As far as I know, pruning for poms is pretty much just about shaping, not so much about production. So I don't think it really matters that much.

Have they started budding out?

Yes, more than budding...... most of the branches have tiny baby leaves.

The pom tree came with the house I bought last year. I moved last October and only saw 2 fruits formed...... not sure if there were more than 2 fruits, which were gone by the time I moved.

No idea how to care for the tree..... or even how old the tree is. It is about 8 ft tall, 4 ft wide with 4-5 main trunks.

Any further tips or info on pom tree care would be greatly appreciated.

Poms will tolerate neglect better than most fruit trees. Specialized pruning is not required. If you live somewhere where it rains, you probably don't even need to water the thing.

I don't really know what kind of fertilization they like... hopefully someone more knowledgeable than I will pop on.

Go ahead and prune now for size, shape, to allow air and light penetration, to remove crossing branches, and to remove water sprouts and suckers. I wouldn't wait too much longer as poms bloom and bear fruit on new wood. I prune mine heavily because they have a will to become bush-like, sucker freely, and need branch and twig thinning for better fruit production/quality.