Most tris have both transitions (swim->bike and bike->run aka T1 and T2) in the same place. So you do set up all of your stuff in one place. Generally you organize it so that your bike stuff is in one pile and the run stuff is in another. That way you just grab your stuff and don't have to think about sorting it during the race.

You come out of the swim, run to your spot and throw on your shoes (and socks if you wear them), grab your bike and helmet and go. At T2 you strip off your helmet and bike shoes, rack your bike, put on your running shoes, and go.

Whether or not you need specific tri clothing depends on how long your race is. If you're doing a sprint, you may be able to get away with regular bike shorts. If the race is longer, then I think you'd be much more comfortable in a pair of tri shorts. There are cheaper ones out there: Aevero and Aerotech Designs make very reasonably priced ones.