A typical TNUA bike is a mountain bike, although TNUAers have shown up on everything from road bikes (fixed gear no less!) to trials rigs. Whatever you ride, you should be comfortable enough on your bike to maneuver it in a group of other riders on sometimes less than ideal surfaces without taking anyone out - including yourself! Your bike doesn't need to be anything fancy, in fact a TNUA would be more fun on an inexpensive hybrid than it would be on a gee-whiz unobtanium road bike with full Campy Record.
The most important thing about your bike is that it is properly maintained before you bring it to a TNUA! Please don't bring a bike that's been in the shed for the last three years and has two flats and a broken derailleur cable. If you aren't sure about your bike's condition then have it checked out by a mechanic before bringing it to a TNUA.
TNUA routes will be on pavement if the trails and parks we frequent aren't either completely dry or completely frozen. Most of the rides we do early in the season will be best suited to a semi-slick or street tire. Once the snow accumulates then a full-on mountain tire will be necessary.
The most objectionable thing about riding in the rain or snow is the stuff that comes up from your tires, not the stuff that comes down from the sky. Fenders are a cheap investment that will change the way you think about riding in the rain! They not only keep you cleaner and dryer, but any rider who ends up behind you will appreciate that you have them too. Fenders come in clip-on and bolt-on varieties. Clip-on fenders are convenient because you can easily switch them between bikes, but bolt-on fenders wrap around more of the wheel and are better at keeping you dry.
Because every frame is a little different, I'd suggest taking your bike with you when you go to the bike shop to look at fenders. Fenders come in different widths, so make sure you get fenders that are wide enough to fit the widest tires that you plan to put on your bike.
At the very least, you need to install a rear flasher! Since all but the first few TNUAs will start out after sunset, you should try to make yourself as visible as possible to motorists. Although TNUAers can see each other fairly well at night, you have to remember that motorists are viewing the world through a windshield that captures glare from streetlights and probably isn't perfectly clean either. Their defroster might not be working and their windshield wipers are probably frozen. You obviously can't see what's going on behind you so you want to be sure that anyone approaching from behind can see you.
You can also put a headlight on your bike if you want to. Most of the time there's enough ambient light to see the ground, but some of the locations we visit are too dark to ride without having at least a few illuminated TNUAers in the pack. A headlight also gives you some reassurance when you're approaching intersections. Reasonable motorists will give you the right of way if they see you, but don't count on them seeing you if you don't have a headlight!
Clipless pedals are the ticket! You don't have to be a "serious" cyclist to notice the benefit from having your shoes attached to your pedals. Clipless pedals enable you to use your hamstrings and hip flexors to pull up on one pedal while your quadriceps and glutes push down on the opposing pedal. This will make you not only a more powerful rider but also a smoother rider.
There are several clipless pedal standards. The most important difference between these standards is the "cleat" that attaches the pedal to the shoe. Shimano SPD and Time ATAC are two common clipless standards that work well and don't jam up with snow and dirt.