You need to play one 1v1 game before 2v2/3v3/5v5 PvP modes become accessible. After that, you need one more battle to unlock the Room feature. This is to familiarize you with the game before you take on other players.
If you have already completed one 1v1 game and you still cannot access other game modes, please check your internet connection. A steady connection is needed for PvP.
As ranked battles are the most competitive battles available in Ace of Arenas, the requirements to join are higher. To be eligible for ranked battles, you must a) reach 800 Credit, b) join or create a clan, and c) own 10 champions (3v3) or 16 champions (5v5).