ASA1, Wecome!
Budget is always a major factor. A lower budget would put you in a springer style rife most likely. I higher budget and interest might put you into a PCP.
Regardless, I'd go with .22 caliber for it's accuracy and knock down power. Different guns like different pellets so, you might have to try a few different ones out to see what your particular rifle likes best. JSB diabolo is generally the most accurate across the board and would be a great place to start.
I'm not too familiar with springers so, I wouldn't be the best to advise on one of those.
PCP's... It's hard to beat the Marauder for starting out and for how quiet it is. But, PCP's require a high pressure air source of some kind. Pumping is an option but, it gets old quickly. The next step is either a bottle or home high pressure compressor. Either way, you're looking at a pretty significant amount of money for air supply. Let's say $600-1000 for a good setup.
If you just want to go with pumping (think "bicycle pump" style) then either the Marauder or Discovery would be my beginners choice. The Marauder fills to 3000psi, the Discovery fills to 2000. It's a lot easier to fill to 2000psi. But, the discovery is not nearly as quiet unless you add a moderator (~$250). Another option might be the Marauder pistol with shoulder stock. This would allow you to shoot with PCP accuracy, and have a smaller air reservoir to fill with the pump. I can't speak to how loud those shoot since I have not owned one.
.177 caliber is an option for rats as well. Personally, I like the added knock down power of the .22 and I find them easier to handle since they're larger. At 50-100 feet, you're going to kill them either way with a well placed shot.
A word of advise, since your used to shooting firearms (We call the powder burners here, or PB's)... Be prepared to start spending more than you thought you would. Pellet guns are not what they used to be and they are VERY addicting!
Tom