Starting issues…where to start?

OK, this is the second time this has happened, so I'll solicit the group on where to start. I'm still learning this car, haven't had it very long.

It's winter here in Chicago, and the car has choke issues, so I try to get it started and running every 2-3 weeks, but it is never happy with me for doing so (have a new carb, just haven't installed yet). I tried to start yesterday, but it wouldn't turn over, so I guessed battery first. I bought a battery charger and battery was low but now...

Starting issues...where to start?