Becoming a bike commuter may be the result of money pressures or whatever, but you'll probably end up enjoying it as a major part of your lifestyle. It's fun and keeps you in shape and in some places and conditions is as fast or almost as fast as driving. And usually more enjoyable.
I'd recommend going to bikeforums.net and reading/posting in the "commuting" forum where tons of people who religiously bike to work will be happy to answer any question you can think of.
Questions:
What kind of bike should I be looking for and how much will I be spending? What other equipment will I need (helmet is obvious)?
A decent bike. Short version...one that's not from Walmart or Target or too old and worn out. You can find single speed bikes on sites like bikesdirect.com that are of quite adequate (though not high) quality for like 300 bucks or so although assembly and hence mechanic skills are needed. Or look around on craigslist. Again, bikeforums.net has people that will help you.
Other equipment - a mini pump (the Topeak Road Morph is a cheap and excellent option; I can't really imagine needing another pump and it costs like 20 bucks), a small kit for basic repairs (fixing flats, adjusting brakes, shifters, and so forth) and lights for riding in the dark. If you have to carry a lot of stuff to work you may want a rear rack that you can strap things onto, or put a basket onto. You can get these real nifty messenger bag/briefcase items that look presentable for the office but also are panniers and attach to a bike's rear rack. I have a cheap one of those and am going to upgrade to a more expensive and durable one when I've got the cash.
A good lock is of course important. Get a solid u-lock - Kryptonite makes good ones. Combination is not as good as key. Keep in mind that bike locks are deterrence, not prevention. Anybody who has power tools can saw through any bike lock; it's a matter of how quick and stealthily they can do it - though it could happen on a busy street if they act like they know what they're doing. Basically, the less protected a bike is, the more tempting of a target for theft it is. Hence, a good lock (or two) makes a potential thief more likely to switch their attention to a bike with a crappy cable lock instead.
But really, only the helmet and lights are strictly necessary, and maybe not even the lights if everything is well lit with streetlights. I never use lights riding around Philadelphia because there are streetlights everywhere.
If you do get lights, don't go cheap - I learned that from experience. You get what you pay for.
Helmets, go cheap if you want. Helmets all legally have to meet certain safety standards, and beyond the basic 20 dollar helmet to protect you, you're basically paying for light weight and aerodynamics.
How long would 6 miles take to bike?
Depends how in shape you, on traffic and hills, and on the bike. It would take me like 30-40 minutes if I were going my normal, not-too-strenous pace.
How difficult would a 6 mile bike ride be physically?
Unless it has some big uphills, not difficult at all - once you get in shape. Also, things like wind and rain can add considerable difficulty. But a six mile commute is really quite short.
Legally, what do I need to know?
All the laws related to cyclists' rights and responsibilities in your state and locality.
Quick tips - you have the right to ride on the road, despite what some drivers may think (and yell at you). Be assertive in taking the lanes that you need when there is not a bike lane.
You are NOT supposed to ride on sidewalks. The exception would be if there's a long, empty sidewalk next to a road with no shoulder or bike lane - then you'd be a bit of a dick not to ride on it, but if pedestrians do come, yield to them as you are the one in the wrong place.
Other than that, the basic rules of driving apply.
Bike pants aren't happening.
I like biking in gym shorts, on a leather (Brooks) saddle. For such a short ride you shouldn't need any special pants, but make dang sure you have a comfortable saddle. I do have a special pair of bike shorts (the right spandex ones with padding) for long hard rides but for riding around town I just wear my normal clothes. When it's warm out I just roll up the right pants leg to keep it from touching the chain and the chain grease. When it's cold I get a rubber band to keep the pants leg away from the chain.
You may get sweaty so it wouldn't be a bad idea to have a change of clothes - have biking clothes plus work clothes. Once you're in good enough shape you may be like me and not break a sweat from a 5-6 mile commute done at about 12-15mph.