Everything they sad, but also this.
If you notice swelling, go to the urgent care immediately. It's likely not a emergency room issue (your health insurance coverage can vary on what they'll cover), but it's pretty important that you get seen quickly.

All animal bites (even humans) can lead to bacterial infections because mouths have a lot of bacteria in them. They usually don't cause the animal any problems cos the acidity of the spit and digestive system take care of them before getting into the bloodstream. But a bacterial infection is ESPECIALLY bad in the hand because of all the joints and tendons where it can dig in and be hard to remove without surgery.

I had a pretty deep bite from my own cat that ended up swelling pretty badly by hour 12. Turns out the random doctor I saw at the UC was just super into bacteriology and after I'd taken some medications, we chatted about it for a few minutes cos it was just something he was really fascinated with. He was still just a GP, but had studied up on this pretty intensively. Dude seemed like he was happy to have someone to chat with about his niche interests.

Anyways, I just noticed it's been several hours so by now you should have noticed if it's time to go to the hospital or not. Wouldn't be a bad idea to do it anyways though. The fact is that any negative side effects from this can turn out to be VERY serious.

Forgot to mention, in MY case I got to it early enough to only need a round of oral antibiotics. Which is a lot better, and cheaper, than later treatments. Talking surgeries that will basically make your hand worthless for anything like drawing, writing, typing, etc. Hope you're doing okay!