"The Decemberists" and "Modest Mouse" are the first indie bands I listened to, and I consider them to be among my favourite bands.
The Decemberists are sort of folk rock, if you want to check them out, personally I'd recommend their album "Picaresque", or if you're feeling ambitious, "The Tain" EP is good, though it's rather lengthy so I wouldn't recommend it until you've heard Picaresque.
Modest Mouse are hard to describe, they're sort of punky music and vocals, but the lyrics are more poetic than the average punk band. Their most recent album "Good News for People Who Like Bad News" is widely considered to be their most accessible, but personally I think that "The Lonesome Crowded West" is a better album.