How do you start in the sport of road biking?

I have a few questions. What are some good beginning distances and speeds for road training if I'm already in good biking shape? What's the best way to meet cyclists and join a cycling club or team? What sort of competitions are good to start trying?

That sounds good. I have been training and I can maintain about 17mph for a good twenty five minutes or so, maybe faster, on a hybrid semi thick tired bike (not a road bike at all though).

Is that really all you can do is just randomly pick a team to join? Is there a better way to find a team?

saying your in good biking shape you should look for approx 20 mile rides ranging from 15mph average on a hilly ride up to 18ish on a more flat ride.... the better you get the more you will want to aim for. a good target is riding for about 2 hours at 17-18 mph. this all depend on the individual tho!
as for how to meet other cyclists.... there are hundreds of road clubs out there so you probably have 2 or 3 right near you.... most clubs will also allow you to enter a few tester races without becoming a member ( just to check you like it)
you got 2 main types of competition.... time trial (which can be pretty boring) or road racing.... as a newcomer you will be able to enter any 4th category road race or any fun races out there without any past experiance!

I just got into road biking for the first time last year with the MS150 and I love it! While the MS150 isn't a competition, it may be a good way to get your "legs" wet. I would first start by finding a hybrid with road tires, or a true road bike if you know you want to break into the sport. Your best resource will be to find a reputable and serious bike shop. They can get you setup and will more than likely have recommendations of local clubs and competitions.

First get a road bike that fits your budget and that isn't a way-old clunker. Then ride it as much as possible. In the mean time, search the internet for local clubs, events, teams etc and find some sort of community or go to a local bike shop and ask where the action is.

Biking shape should be determined by Heart Rate and if you can afford it, power. Avg speed doesn't really mean anything unless you are on the track where the environment is more controlled. I mean I oculd do the same road one day and with a tailwind and me feeling fresh avg 25mph but the next with a severe headwind and avg 15 or 17 even though the effort was the same.

Just look for cycling clubs in your region (usa cycling has a good resource, if your from the usa) and then go to their pages and pick one that you like (one with lots of rides, training races if you want, etc.)

The people you meet as a result of joining a club and going on local rides (often offered by bike shops) should be more experienced and answer any other questions.