Regarding general indie music, I figure you should start out by picking up a couple of the generally well-regarded albums, and so I'll try to keep away from personal favorites.
Neutral Milk Hotel - In the Aeroplane Over the Sea
Folky, meandering, and very lyrical - this is somewhat of a love/hate album, but most people I know fall in the love category.
Built to Spill - There's Nothing Wrong With Love
If you like more jammy, guitar-based stuff, go with Perfect From Now On (my favorite from their catalogue), but both albums are excellent - a collection of truly unique and awesome poppy, rocky, intelligent tunes.
Radiohead - OK Computer
I don't know anyone who doesn't like this album - sometimes heavy, often spacy, and always something new.
The Clash - London Calling
Okay, not a "new" or "indie" record by any means, but if you don't have this, pick it up - a true classic.
The Dismemberment Plan - Emergency & I
Very angular and yet very melodic, this album has a little bit of everything, but is in general split between angry and slow songs, and full of excellent musicianship.
Pavement - Crooked Rain, Crooked Rain
I find this more accessible than Slanted and Enchanted (their first, most influential record). One of the first bands that fit neatly into the most commonly labeled "indie" genre, Pavement's albums are full of jangly pop lightly sprinkled with a number of influences. Crooked Rain has their near-single "Cut Your Hair", as well as Range Life, Gold Soundz, and the epic closer Fillmore Jive.
The Wrens - Meadowlands
This is a sad, powerful album, with heavy guitars and heavier lyrics. If you like your music thick and lasting, check out the Meadowlands.
That's it for now, but all these albums are considered highly within the indie catalogue (well, and then there's London Calling), and have had time to settle into a definite amount of praise and credibility. Hope I helped!
