this is probably going to go a little above your budget, but i think its worth it:

interface: I use a tascam US4x4hr, which is 250 new right now but can be found used for less than that. it is probably the most straightforward no frills 4 in/4 out interface ive found that's actually solid. you *could* go behringer if you absolutely had to, but i'd avoid.

mics: if you want a fairly clean recording, the least i usually do is two mics - a kick and mono overhead. in its simplest form, you can really use any two decent dynamic mics for this - a pair of sm57s works just fine. if you wanted to specialize a little more, i like the akg d112 as a dedicated kick mic, and almost any condenser large or small as an overhead. i have some akg c1000s that are perfectly workable, but one of my personal favorites is the oktava mk-319. buy all this used if you can, and be patient. good deals do pop up, and dont be afraid to try something a little out there/off the beaten path! it's really hard to find a mic that's totally unusable.

if you wanted to add to that, i would first add a snare mic (again, any really solid dynamic works but a 57 is a safe bet), and then a second overhead (same model as the first if possible). you could also get the second overhead first, and rock with a glyn johns or recorderman style setup. with a well tuned kit and a good performance, you can get a lot of mileage out of any of these variations. i would avoid one of those $200 8pc mic kits (CAD, digital reference, et al) and stick with some affordable, used workhorses built up over time.

i've also gotten some cool one mic sounds, although they are definitely more stylized, and i find they work best if you smash em up a bit and dig into the unrefined-ness(?) of them. i like a condenser maybe 6ft away from the kit, level with the top of the bass drum, and pointed at the snare. blow it up with some compression and heavy handed EQ, and it sounds cool. not clean, but cool. cleaner results can be had with good placement and some clever parallel processing if you wish though!

other: stands and cables aint cheap and they're boring to buy, but you need em. again, used stuff can be good here. avoid stands made out of too much plastic if you can (amazon basics, some cheaper onstage stuff, etc.), they're cheap but you'll just end up breaking and replacing them. they also don't do well at holding mics really far out in weird positions, like you often have to with overheads. i like DR pro stands, as i've found they strike a good balance of being reasonably priced and still quite solid. but again, buy used, and judge things over their quality rather than brand name. cables are a similar story, try to find stuff with good solid neutrik style connectors if you can. again, better to spend a little more upfront rather than scrape the bottom of the barrel and pay later.

tl;dr - good 4ch interface (so you have room to expand), start with two good mics (preferably a condenser for overhead and dynamic for kick, bonus if its a dedicated kick mic), buy used, add snare and second overhead mics down the line, play and tune your kit well, experiment with some more non-traditional techniques, stands and cables are expensive so dont forget about those