Good advice there from Musiclab. Get the overheads as right as you possibly can first, then move on. If you are in a room that's not well treated, your job is more difficult by far. Even temporary, makeshift diffusion and absorption can make a difference on drums because their high transient response makes for a ton of early reflections in an untreated room, especially if it's a smaller room.
In terms of avoiding bleed, have you listened to the effect of snare mic position and angle on the bleed? Take your time with this. Likewise on EQ, compression and gating, take the time to wring it out. The individual snare and kick tracks, once optimized, might not sound so good on their own but in the mix, they work.
You can also make a kick tunnel pretty easily to help isolate the kick from the snare. I've used tables and blankets as a DIY tunnel.