I do  .  Before you decide which bike you want to buy, here are a couple of questions:

1. How much are you willing to spend?
2. Do you want to ride off road at all or just on the road?
3. How many miles do you think you will end up riding? How far is your work?

If you are just going to be riding on the road or need to go more than 2-3 miles, I definitely would recommend a road bike (skinny tires, drop handlebars).  They are much faster and lighter as long as you are going to be on the road and if you have to go very far it will be a lot faster on a road bike.  If you are on a limited budget you probably pick up a decent used one for a few hundred bucks or for a few hundred more a decent new one.  I would definitely steer away from department stores bikes especially if they have any type of suspension system.  These bikes are very heavy and you will have to do a lot of maintenance on them.  Besides if you look hard enough you can find a good used bike for the same price.  If you want to ride off road and you can only buy one bike, then buy a mountain bike.  If you are buying a new mountain bike, unless you are spending more than $700-800 don't waste your money on a bike with rear suspension or disc brakes, because the bike will have cheaper, crappier other parts to make up for the cost of those.  A big drawback to a mountain bike is if you have to ride very far, it will be significantly slower than on a good road bike.  For example on a flat stretch I can easily hold 22-23 mph on my road bike, but about 17-18 mph on my mountain bike. That doesn't matter much if your work is only a couple miles away, but makes a big difference if you work is far away like mine (20+ miles)

I would suggest a a decent mountain bike. From my experience they hold up best to long term day-to-day rides.

For your price range there a couple "fringes" you might want to look out for.  If they try to sell you on it have some bike because it has something or other made of carbon fiber, it is just a gimmick, you will get crappy carbon fiber quality and lose money you could have spent to get better quality parts.  They also might want to sell you on some fancy wheels.  Again for your price range you just want good solid parts, though if you are on the bigger side (like me around 200 lbs), you may want to invest is more durable wheels as the stock wheels are usually suited for people more in 150-160s range. If they try to sell you anything that automatically shifts for you it is a piece of crap. Also don't go in for a wide super soft seat.  It might be comfy on the show room floor, but if you ride very much you will get major chaffing.
Some things you want to consider that they will probably suggest is "clipless" pedals and shoes to go with them.  If you are riding 15 miles a day to work and back you will not regret it.  Also get a decent helmet, one with lots of vents.

Something else to consider is when they fit you for your bike they will probably ask you how upright of a position you like to ride in.  Most people who don't bike very much while opt for a more upright position, because going on a ride around the block that is more comfortable, but if you are going to ride 15 miles a day, you might opt for a less up right position because this puts more weight on your hands which makes it so road vibrations and bumps don't go straight up your spine.  It is also more aerodynamic so you go faster. It takes some getting used to and if you have week core muscles and arms it will tire you out at first, but you have been doing parkour for a while so you will probably be fine.

Chris, is "derailleur" pronounced the way it looks like it should be pronounced in French?  Or do I say that with a typical American accent?

I have always heard it said dee-rail-er.  If you came into a bike shop and said day-rye-euh they problem would either think you were a total novice and try to sell you some chitzy hybrid bike or if they were totally bike geeks who have on video the last 50 years of the Tour de France they might think you were so cool for saying it in French that they would try to convince you to buy $5000 15 lbs super race bike.

Question...  I might go used, but if I do...  I'm about 5'7".. what size frame should I be looking for, ideally?