Hi,
you'll be fine with the mountain bike - just take off anything you don't need for a road race (mudguards, lights). The minimum you need for a triathlon is stuff you already have - swimsuit, bike, trainers, shorts & shirt. You can pull short & shirt on after the swim. (leave the swimsuit on....no nudity permitted!) You have to dry off first though, or getting clothes on is ridiculously hard!!
If you're happy to spend a bit, then a trisuit is a reasonable purchase. You swim, bike & run in it - no changing needed. A tri-suit will have a small amount of padding to protect you on the bike - not as much as regular bike shorts, but enough, and it dries much quicker than bike-short padding would.
Another good (and cheap) purchase is a number belt. It's an elastic strap that you attach your number to. Then when you come out from the swim, you pull the belt on....number at the back for the bike, twist it round to the front for the run. If you don't have one, then you have to either pull on a shirt with the number pinned on, or spend time in transition pinning your number to your tri-suit.....
How it comes together?
- register
- rack bike (means put it in the transition area, with your trainers & anything else you need for the bike/run)
- swim a bit
- head to transition (the place where bikes are stored)
- dump goggles, hat, stand on a towel to clean/dry bottom of feet, pull on socks/shoes/number belt
- put on helmet BEFORE you take the bike off rack (DQ if you don't have helmet on and fastened when bike is loose)
- bike a bit
- back to transition
- bike back on rack
- take off helmet
- change shoes if you use cleats on the bike~
- number belt twisted number to front
- run a bit
- feel chuffed that you've finished your first tri
- go home, enter another one and spend the rest of the weekend perusing bike & kit websites
Enjoy 
