Solved!

I'm wondering about right now is: How do I feed my pets? Some of my pets are "Hungry" and I'm not sure how to fix that.

Also, so far the only way I know that I get traps is through encountering Sylesti in the Festival explore zone; someone correct me if this is wrong.

To feed your pets, you can do it one of two possible ways.
You can click on the pet and scroll down. There will be a button 'Feed Sylesti'. Click on that and then pay to feed the pet.
OR
At the top of your Stables tab, there should be a few buttons at the top. Among them 'Feed Pets'. Click on that and then click on one of the options 'Current Stable' or 'All Stables' according to which one you want to do.

I have also heard you can hire a stable hand - one of the blue things in the top left corner of the screen when you have reached the stables - and the help will feed them. I haven't tested that, though, and have no experience through that.

Hope that helps with the food problem.

on stablehands, once you have achieved 10k AP (you just have to reach it once and it unlocks forever regardless of AP you have on hand) you can click the "hire a stablehand" option.
You can then pay for Rosemary to automatically feed your pets every day for a small cost (2 weeks is 1400g, more than two weeks starts costing diamonds as well, but I can't check the exact prices since my contract with her is still going)
Not much point when you only have a small number of pets, but once you start getting up in numbers it is worth it price-wise (I think someone did the math once but that was a while ago)

Ah, got it! Thanks for the help. I don't think I'll be using a stablehand yet. As you said, Scathreoite, only four pets! But maybe after I've got a lot, possibly.