First Ever Tri - What kind of Shoes do I wear?

Welcome to the sport!  Lots of us have found over time that we prefer to have cycling specific shoes that clip into the pedal, and therefore have to change shoes.  However, this is not necessary.  You could wear your running shoes with regular pedals.  If you want to go that route, ask at the bike shop about "toe clips" for your pedals.  These will hold your running shoe in place, but allow you to jump out and run also.

However, if you really like biking and plan to do more, you may wish to investigate cycling shoes and "clipless" pedals - which you "clip into".  There are many opinions around here on the best type/brand, although I would recommend a trip to the bike shop.  It's easiest if they help you and will ask questions about the type of biking you're planning to do, etc. 

Good luck!

By the way, I would heartily recommend asking about cycling shoes made specifically for triathlon - if you do investigate clipless pedals.  I know many people who started with standard road racing shoes and then ended up needing to buy tri specific shoes.   Their advantage is you can wear them without socks (good for wet feet) and they have a single strap, which makes the shoe change much simpler.  They are just as comfortable and useful for riding, I think even more so.  Just an fyi.