I signed up for my first triathlon this coming October.  I've got under three months to get in shape for it...  

Three months is a little short notice... isn't it? I knew a guy who did it on a whim like that, but he was already a crew team member and in fantastic shape.

I dunno, I am not exactly a couch potato.  
Besides, it's my first one...I don't have to win, just finish.  I know I can do the run and the bike, my only concern is the swim.  I've only been swimming for a bit over a month now but Friday I was able to make 20 laps, 1000 yards, without stopping for rest.  Gonna be a bit different in a lake instead of a pool, but then again it's only a quarter mile swim.

Which triathlon? Is it a sprint or international distance? Or HIM or IM?

It's a sprint––-I don't feel comfortable entering a race with a swim longer than that just yet.

I am going to need to buy a road bike, I guess.  All I have right now is a mountain bike.

I checked out road bikes today...holy SHIT!  $800 for the cheapest one they had!

look for used ones. you dont need anything special as a amateur and unless you plan on doing the half ironman or full ironman, i would save thee $ and buy used until you realize biking is your thing or not.
good luck with your training. 3 months is PLENTY of time to train for a sprint, unless you have health problems or are a huge couch potato.
even then, in 3 months of serious training EVERY day, you should be able to accomplish enough to at least be confident in your ability to finish with a respectable time.
time really does not matter, finishing and having a good time during the event and training for it are what it is all about!
why do it if we dont enjoy it?

I have been doing a lot of mountain biking (well, this is Florida, so let's call it "trail biking") and I've really enjoyed it––-I think I would like to give road biking a try.  I've been looking for used ones, but the only bikes I've been finding on craig'slist that are worth a shit are mountain bikes.  
A friend of mine who is a big time triathlete––-he's about to do an Ironman in a couple weeks––-has offered to loan me his spare road bike for the race, but I want to get some practice on whatever I am going to race on.

I rode my mountain bike on the road/bike path for 20 miles yesterday up some pretty fierce hills (for Florida ) and it wasn't much of a problem, other than my ass hurt like hell afterward.
Maybe I'll just do the triathlon on the mountain bike.

The hardest part IME is the swimming in a mob of people in the ocean or a lake. Nothing like getting kicked in the head and slapped on the back by 300 strangers.

The hardest part for me will be the swimming without the drowning part...