I’ll go ahead and echo a lot of the great suggestions listed here already (as they’re some of my favorites bands/artists) and add some of my own. I apologize if these bleed between the lines of true indie rock and alternative rock (these labels are nearly meaningless to me).
Belle & Sebastian - British pop, very melodic, very soft. I’ve heard them described numerous times as “precious.” Some of the lyrics are just so sweet. Might take a few lessons to sink into, but it’s worth it. *If You’re Feeling Sinister *and Dear Catastrophe Waitress are my favorite albums.
Joy Division - Dark, tortured, depressing lyrics. A short lived band that sort of bridged the gap between synth and punk. After the lead singer’s suicide, the remainder of the band went on to form the great 80s group New Order. I’d recommend Substance (a compilation) to new listeners as it was my first introduction to the band. Their two official releases, Unknown Pleasures and Closer, are some of the finest albums of the period.
My Bloody Valentine - As stated, Loveless is one of the greatest albums of the 90s. It’s the crowning achievement of the “shoegazer” movement. There is a blending of lyrics and distorted guitars that is unparalleled and it works perfectly. This is one of my Desert Island Albums. Isn’t Anything is an excellent album as well. In this same vein, check out Ride’s album Nowhere. Lyrics are easier to follow if you’re into that.
Pixies - Sort of an infusion of surf guitar and punk. Great, catchy, short tunes. Lyrics are crazy and all over the place, but they work well within the context. Surfer Rosa and Doolittle are probably their finest albums, but I love Bossanova and Trompe le Monde equally as much. They were one of Kurt Cobain’s biggest influences and are college rock legends.
Sonic Youth - Noise, distortion, feedback, experimentalist rock. Daydream Nation is regarded as their best and most influential of many excellent albums. Dirty has more of a punkish or grunge feel to it. Murray Street and Sonic Nurse, two of their newest albums, are more melodic and I’d highly recommend them to new listeners who might be turned off to the initial “noise” of some of their earlier albums.
The Strokes - Is This It? is solid from start to finish. Very catchy with good hooks. One of my favorite recent rock releases.
Yo La Tengo - I Can Hear the Heart Beating As One has sort of a subdued feel to it. Melodic and the occasional fuzzy guitar. I’m just getting into these guys, but like them very much.
Also, if you haven’t, check out The Velvet Underground. They’re one of the biggest influences of indie/alternative/whateverthehellyouwannacallit.