Most of the mixing, etc., for producing is done on the computer. IMO the point is to get the best quality audio into the computer first and do the rest in software. So I wouldn't worry about a physical mixer. Unless you need one for live performance or something, but even then you can probably use a laptop.
Would you be recording these tracks all at the same time or separately? The Focusrite Scarlett series is really nice, and the preamps sound great. They have several options between $150 and $350 (2i2, 6i6, 18i8, etc.) with different numbers of inputs. I have a 2i2.
I do mostly small ensemble woodwind recording, and I use a pair of Audio Technica AT2020s for that purpose. There are other recommended mics for different applications. The Sure SM57 is a pretty popular vocals mic. Not sure about the best mic setup for drums or guitar though. (I've worked with some people who were recording guitar, and they usually just put a single mic on a stand about 18" away from their guitar and pointed it at the hole in the body. Acoustic guitar, obviously. Electric you'd just run a line to the audio interface, or use an amp and run a line out from that.)
Most of your money will probably be spent on mics. IMO the only way to really do it "right" it to try a bunch and see which ones give you the sound you want. *shrug*
