Your local building permit office will tell you all this. I received a
hardcopy guide from my county intended specifically for homeowners
finishing their own basements. It was quite handy.
Basically, you'll want 2x4 framed walls, R11+ insulation, and a
continuous vapor barrier "on the warm side of the wall".
Even if the codes allow furred walls, don't. Just don't. I can send
you photos of what it'll look like in 35 years.
There are codes on the unfinished space. The under side of stairs must
be drywalled and each room must have a light. There should be an
outlet near the furnace. It's all in the code book.