Many people use a race bike for fitness, but you may find that you want to use the bike for shopping, commuting, trips to the gym etc, so a bit of practicality comes in handy (fittings for rack and fenders).
Old sport bikes are really good general purpose hacks. You can pick them up used for $50 and use them to figure out what kind of riding you prefer, how you like the bike setup and what size you need. You may have to invest in new tyres/chain.
Suggest you find a good local bike shop and get to know them.
If you want something new, then Fuji do a nice line in good sport bikes at reasonable prices such as the League or Ace model.
Budget for shorts, helmet, gloves and lock.