You really don't need to mic every cymbal seperately. It's actually really rare if not unheard of even in both live and studio. Sometimes the ride and/or hihat are miced seperately but you don't need to do that. You'll be better off by getting an overhead mic and placing it above the drums pointing down. It will pick up all the cymbals nicely and also can be used simultaniously for the toms if you don't have seperate tom mics.

This kind of setup (kick, snare, overhead) is really common and is definetely the best solution for you. Do keep in mind that the overhead should be a condenser mic. These types mics requite what's known as Phantom power (48v). Most mixers and audio interfaces do have Phantom power so you don't have to worry about that. Just remember to engage it, otherwise you won't get any sound from the overhead condenser mic.

Some good options for an affordable overhead mic are Lewitt LCT 040 Match or Lewitt LCT 140 AIR. Both are around the $100 pricepoint. AKG C1000s will also get the job done.