So I just became a LT. Commander, I got to chosse category of ship, Science, for 1 $. However after I claim my ship and I go back to the girl upstairs and in her store I see I can still buy a categorie of ship for 20,550$ What the heck? If I buy again the Science class will it be Tier3? I it cost me money to buy let's say Cruiser or Escort class?
Nope, you just get that first ship per tier free, again at cmd, etc. The ones you pay for will be the same "level" of ship as you get for free. The only reason to pay is to get one of a different class than you got for free. If you pay for another science ship (no reason to really) it will be basically the same as the one you got for free. Although you can customize it differently than the first one, there is really no reason to do so with the same type of ship.
edit: to clarify, you get a token on ranking up, which you can pick one the three, doesn't matter which, it's basically "free". If you want a different type after that, yes, it will cost you. That cost seems a lot, but it's not as bad as it looks.
Read the description (or next time you reach a new rank anyway).
You didn't buy the ship for 1 energy credit. The admiral gave you a starship plaque or something like that (server's down right now, can't check) which acts like its own currency that basically lets you get 1 free ship for whatever rank you've just reached. When you hit Commander/Captain/Admiral, it'll be the same thing.