What does Peeta like about himself?

I would say that the times when Peeta Mellark likes himself best is when he is helping to keep Katniss Everdeen alive.
Peeta isn't a boy with much self-confidence—this is seen early on in the first book in the Hunger Games trilogy, when he tells Katniss that while she might win the games, he has no chance whatsoever. This is despite the immense strength that he displayed from all those bags of flour he had spent his life carrying as a baker's son.
It is made clear at the beginning of the second book that Peeta would never have forgiven himself if he had allowed Katniss to go back into the arena without him—again implying that he likes himself best when he is protecting the woman he loves.