£350..not a lot but it's a good start to getting MORE than an "alright" drum recording. You have PLENTY of power with your PC so no worries there means more money to spend on equipment.
I'd go the mixer and A/D converter card route giving me more room to expand and add to my setup as time goes on.
Off the top of my head...
Mixer:
Yamaha MG124C retail at £149.99 you could pick one up on ebay for £100 or less. This will give you 6 mic pres with phantom power, EQ, Insert points (you can use these to feed your audio card), headphone output, monitor output etc. It's a nice little mixer and its made by Yamaha, good reputable company with good customer support and waranty.
Soundcard: M-Audio Delta 1010LT: around £120 new, they go used all the time on ebay for around £50. This will give you 8 inputs into your computer so there is room for future expansion or for recording other members of the band at the same time as your drums. It's a 24bit 96KHz 8in/8out audio card with the bonus of midi (whether you need it or not.). The 1010LT has 2 built in mic pre's without phantom power and 6 line inputs. M-Audio have been around for a few years now and have good customer service, good solid drivers.
If you couple your mixer and soundcard you can use the 6 mic pres on the mixer connected to the 6 line inputs on the 1010LT and 2 line outputs of the mixer to the two mic pres on the Delta card or 2 mics directly into these...whatever way you work it out you have the ability to record 8 different mic signals on seperate channels on your computer....4 for drums, bass, guitar, keyboard & vocals...all at the same time. With each element of the band on its own track you have a LOT more control over how good you can make everything sound.
Mics: you say you need SDC's for overheads. Someone mentioned
the behri C2's..... I say
the AKG Perception 150. AKG have a new line out so these have taken a HUGE fall in price recently.
www.dv247.com are knocking these out in pairs for £79 quid. I've used them on drum overheads, acoustic guitar, fiddle, piano, a choir...all with great results, they are great little mics.
That comes to £340 if you buy everything brand new but if you do your shopping on ebay for the mixer and Delta you've only spent £230. Leaves you £120 for a drum mic and all the cables and stands you'll need.
Using a mixer will eliminate any latency in your monitoring which can be very putting off, especialy for a drummer. You say you have some "cheapo" other mics so you can use these on other instruments to get an initial recording so the whole band is playing together.
That's my first thoughts on your budget for what you want to achieve. No doubt someone will chime in with a one-box-does-all wonder that has slipped my mind
You're young, you're starting out, so read everything you can and build yourself up a list of possible routes and equipment choices and save till you can afford what you need. Refrain from the temptation to buy the cheapest you can to get your job done because ANYONE who has started out on this path knows if you buy cheaply, you shall surely buy twice. Buy quality and it will last you till you outgrow it...and it'll have held it's price well when you want to sell it on.
Good luck with it,
Alex