I want to further expand my tastes, so I thought I’d try some of these two genres. Open to any suggestions 
Sorc and I can help out with the rap. I don’t really know anything about the blues though.
Here are a few artist you should check out:
Ludacris - He’s got a lot of albums. Pick up the latest one, and if you like that, definitely check out his older cds.
Timbaland & Magoo - Timbo is a producer who also raps. He makes great beats. His albums with Magoo are really good. Check out Under Construction Part 2 or Indecent Proposal.
Mobb Deep - They’ve been around forever. They are currently in G-Unit. All of their cds are good.
N.E.R.D. - Technically not rap. It’s The Neptunes plus one other guy. Sort of a rock band made up of hip hop people. Very very good.
Then of course there are the classics. Dr. Dre, Snoop, Tupac, Biggie, Wu Tang.
Do you want gangsta rap, sit back and relax rap or hip hop?
As far as blues, you can’t go wrong with some B.B. King.
Blues, eh? You should check out some of these guys:
B.B. King
Robert Johnson
Elmore James
Booker T and the MGs(something like that)
Stevie Ray Vaughan
if you want to expand your musical taste, listen to the pixies and the velvet underground. then find other bands like those.
I love Wu-Tang Clan, for rap. I’ve only heard their first (Enter The Wu: 36 Chambers) and second (Wu-Forever) albums, though. But I really like them.
Hieroglyphics, A Tribe Called Quest, Souls of Mischief (especially 93 til Infinity), there’s a local group called The Team around here (some good songs: Just Go, Show Me Your Nasty, It’s Gettin Hot), Keak da Sneak (esp. Super Hyphy), Ride by EA Ski, etc. etc.
Well, if one wants to expand their tastes, one shouldn’t conform to just one set type of music. Maybe he’ll look for more independent stuff later on, but right now this is a thread asking about blues and rap (although, on the record, The Pixies and The Velvet Underground do kick ass).
Rap:
NWA, Killah Priest, and Gravediggaz are the only ones I really have thought were good.
Blues:
You can never go wrong with Howlin’ Wolf, Robert Johnson, or B.B. King (as has been stated).
George Thorogood and the Destroyers are Bluesy rock, and alot of his stuff is covers of older blues songs. The first Shinedown album was pretty good, sorta what you’re looking for.
And seconding what GSA suggested blues-wise.
For rap, since you didn’t answer, in general I would go with Tupac, Notorious BIG, Mobb Deep, The Game, Eminem, Lil John, Lil Wayne, T.I., Ludacris, Young Jeezy, Nas, Bone Thugs n Harmony… I’m sure I can come up with more.
For rap, I would recommend Necro. But that’s just me.
Thanks all. And Sorc, I didn’t answer because I’ve been sick as shit. Sorry, yo.