No, it doesn't sadly. Until level 16 (When it evolves)does it learn a new move. The new move Abra learns is Confusion after it evolves.
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Abra doesn't learn any attack moves until it evolves at level 16 into Kadabra when it learns Confusion. You can teach Abra some TM's if you need it to learn an attack move sooner.
The only move that Abra can learn is teleport, however you can teach him TMs. When it evolves in Kadabra at level 16, it learns its first move: Confusion.
when it evolves (lv.16 )(Confusion)
Abra doesn't learn any other moves, but when it evolves into Kadabra at level 16 it learns Confusion.
Your Meowth will evolve into Persian when it reaches the Level 28. Remember that only Meowth can learn the move "Pay Day", so wait until it learn this before evolving.