I'm using some cheap (25€) dynamic behringer microphones with cheap preamps, they sound absolutely horrible for singers but go nice for micing distorted guitar amps. For the singer I use a good AKG.Emotion series microphone C900 with a decent preamp with some singer-specific options in it. It's a condenser microphone and it sounds absolutely brilliant (it should because I paid over 300€ for the combination).
Actually if you want to stream you can manage it with two decent condenser microphones, you need two preamps with phantom power for them and don't need a mixing desc. The drawback being that you won't be able to adjust anything.
The best method is, according to me
, putting a microphone for every instrument and singer. If the guitar amps have line-outs you can use that and save on micropones. Then you let everybody adjust their volume so the band likes what they hear, after that you adjust the faders on your mixing desc and listen to the headphones and make sure that sounds good because that is what people will here when you are streaming it on the net.
If you want to be sure you can record a piece first and then play it back, if that sounds good it's OK and you can start streaming.
If you want it to sound good you need good equipment and that costs money, but when I started I had almost no equipment at all and I managed allright, you might find some friends with a microphone or a mixing desc lying around, so use those before you start buying stuff.