clothing for training?

For your next tri, I'd get tri shorts.  You can do the whole race in them and they'll probably be more comfortable on the bike than your bathing suit.  A tri top might be a good idea too, unless you're going shirtless (but some races don't allow that).

If you're in the DC area, I'd invest in some winter cycling clothing if you plan on cycling outside.  Maybe someone who lives in a cold area can chime in on that.

There's nothing wrong with cycling/running in cotton if it works for you.  I prefer cycling jerseys because they hold less moisture, have pockets for my stuff, and come in bright colors (I think visibility is a good thing).  I would invest in cycling shorts if you plan on doing long rides, since chafing is one of the last things you want to deal with when you're out riding.  Running shorts might be a good idea, since they tend to be lighter and more comfortable for running than other kinds of shorts.

For your next tri, I'd get tri shorts.  You can do the whole race in them and they'll probably be more comfortable on the bike than your bathing suit.  A tri top might be a good idea too, unless you're going shirtless (but some races don't allow that).

If you're in the DC area, I'd invest in some winter cycling clothing if you plan on cycling outside.  Maybe someone who lives in a cold area can chime in on that.

There's nothing wrong with cycling/running in cotton if it works for you.  I prefer cycling jerseys because they hold less moisture, have pockets for my stuff, and come in bright colors (I think visibility is a good thing).  I would invest in cycling shorts if you plan on doing long rides, since chafing is one of the last things you want to deal with when you're out riding.  Running shorts might be a good idea, since they tend to be lighter and more comfortable for running than other kinds of shorts.

I would invest in a nice pair of bike shorts for training. You can find some decent ones at Target and Performance Bike pretty cheap.