I do a lot of container gardening. For shade or partial shade, I love torenia--it goes all summer with little care. For sun you can always go with the old stand bys (they're stand bys for a reason!) like petunias--try the 'wave' variety, or geraniums. I like snapdragons in pots, too, and they keep blooming into cool weather.
Those are all annuals; I don't usually put perennials in pots, but have grown hostas (shade) and grasses (sun) in pots. Just try whatever interests you--the annuals aren't too expensive and you can have a lot of fun experimenting.