Do we need a producer?
Yes and no. Your band should be the most important producer; after all, it's your music. However, it's really handy to have another pair of ears around that's NOT emotionally involved. To me, a producer is a lot like a proofreader and editor. They take what you do, tweak it a little or a lot, point out mistakes you might not have noticed, and put it all together.
See production page

What songs should we record?
Record your best songs. It's important to make a distinction between good live songs and good recorded songs. some of the most powerful live music doesn't translate well to a CD without the audience to feed off of.
It's better to record fewer, better songs than to record more, crappier songs. You may not want to admit it, but we ALL write crappy songs (some more than others).
Record songs that are fun to play. You're probably gonna end up playing them a lot.

What gear should we take?
As little as you can get away with and still get your sound.
For guitarists, I'd suggest an amp (head and cab), a guitar, your basic effects, and enough cables to put everything together. It's been my experience that studios have spares for most stuff that can fail. They probably also have better effects than your Crap-o-Tron flanger, too. Bring spare tubes, though.
Bass, for a demo project, is usually recorded directly into the board, so there's no need to bring an amp.
Drums can sometimes be provided by the studio, unless you're so totally in love with your kit that you can't play anything else.
Singers should bring a big bottle of whisky. (Just kidding.)

How should we prepare?
The most important preparation to do is of the music. You're paying for studio time; it's cheaper to make sure everything's ready beforehand. Practice time is free, or close to it. Use it. Practice the songs over and over, until everyone could play their part backwards and in their sleep.
If there are going to be any changes to the arrangement, sounds, or lyrics, now is the time to make them.
Get your gear in shape. New strings, drum heads, tubes, or anything else you might need. Get the guitars and bass set up. Get the piano tuned.

What is the mixing process?
Basically, you sit and listen while an engineer tries a bunch of different things with the mix.
It's probably a good idea to let the mixing engineer work on his or her own for the first hour and get the basic sounds together.
A song is mixed the same way it was recorded; drums first, then bass, then everything else, with vocals last. Each track gets its particular sound down, effects are added at the board, and general levels are set. Once the sound of each track is dialed in, the whole song is run through to listen for any places where special mixing attention is needed.

Once everything's set, the mixing engineer does a "dry run", running the entire song through with all the settings and all the tweaks. Finally, the DAT is positioned and the engineer runs it for the final mix.

Should we master it?
Yes, unless you REALLY can't afford it. Mastering makes a difference that's not so much audible but emotional. Well, it's audible, too, but it's only really noticeable in its absence. Remember that every CD you see on your shelf is mastered.