Sonic Youth is my favorite band, ever. I'll start off with a few random ramblings so you can know a little about them before I get into recommendations etc.
They've been making music for around 20 years, starting out in New York City. Their principal members are:
Thurston Moore: Guitarist, and sings on a lot (don't want to say 'most') of their songs.
Kim Gordon: Bassist, sings on most of the songs that Thurston doesn't sing on. It seems a lot of people dislike her voice - it can sound grating sometimes, but other times can be beautiful (see 'The Sprawl').
Lee Ranaldo - Guitarist. Sings on the songs that Thurston and Kim don't sing ong. This usually comes to only a few songs per album.
Steve Shelley - Drummer. Doesn't sing on any songs that I can recall.
Jim O'Rourke - New member of the band (compared to the other 4, who have been together since 'EVOL'). Does a lot of stuff and has worked with groups like Gastr del Sol.
I'd say that the most famous (infamous?) aspect of Sonic Youth's music is the noise. Not without reason - a lot of their songs have extended portions where it sounds like they're just playing as loud as they can and creating a huge distorted mess. But eventually you come to appreciate these 'noise-fests' and view them as a very important part of the music. This isn't to say all of their stuff is like this; they are capable of producing quite the melodic tune (see 'Sugar Kane'). So as long as you keep an open mind with regard to this kind of thing, I think you'll find Sonic Youth's music to be infinitely rewarding...
OK, now on to some comments regarding albums/songs. I don't want to write a novel here, so I'll stick to the ones I know/like the best:
Daydream Nation - I had to start with this one. It's my favorite of their albums, and a lot of people consider it their masterpiece. There isn't a bad song on it - 'Teenage Riot' is incredibly catchy, 'Silver Rocket' rocks HARD (and contains some of their most abrasive - and in my opinion, cool sounding - noise), 'The Sprawl' is probably my favorite Sonic Youth song ever, and even 'Providence', which is bizarre to say the least, has a great piano thing and... well, it's just neat. I can't recommend this album more.
Dirty - This album was made during the grunge explosion, and Butch Vig produced it. Far from being a "sell-out", though, this album contains some of my favorite SY songs. 'Wish Fulfilllment' in particular - a Lee song that must be heard to be believed. Other great tracks are the aforementioned 'Sugar Kane', 'Swimsuit Issue' (seething!), 'Theresa's Sound-World' (gorgeous), 'JC' (spooky!), and 'Youth Against Fascism' (political). This has 15 songs on it, so you get a lot of SY for your money. Also, this was re-released in a 'deluxe' edition a month or two ago, with an extra disc with a ton of new stuff. I'm planning to buy it soon.
Sister - I think this is Salome's favorite album, so I'll let him comment on it

. I'll just say that 'Schizophrenia' is musical perfection and the middle section of 'Pacific Coast Highway' is the coolest thing ever recorded.
Washing Machine - This album is from the mid-1990s, and it's, in a word, awesome. The opening track, 'Becuz', blew me away when I first heard it. Same with the title track. And this album ends with 'The Diamond Sea', a 20-minute epic with poetic lyrics and enough beatiful noise to last a lifetime. It has to be one of the best SY songs ever.
Murray Street - Their newest album, released last year. 'The Empty Page' rocks. 'Disconnetion Notice' is great. 'Karen Revisited/Karenology' is awesome. But 'Rain on Tin' is AMAZING. Has to be one of my top 5 SY songs.
I'm tired from all this typing, so hopefully that will give you a good start on Sonic Youth and all their musical goodness

. I'll try my best to answer any questions you have!
Here's to the begining of your Sonic Youth odyssey...
