Huge, wide-ranging genres to recommend from! Funk sounds range from R&B, soul and jazz, through disco and on to rock and beyond. Some earlier artists that come to mind:
The Meters - The Meters is a must for any funk collection - one of the earliest influential examples from New Orleans. Have a listen to Cissy Strut and Live Wire.
Funkadelic's Maggot Brain is an essential. Connected with Parliament and George Clinton. Funk with touches of psychedelia, it's an awesome album. Similar to Funkadelic, but later moving more towards disco and rock is Sly & The Family Stone (The Meters covered their Sing a Simple Song).
Booker T. and the MGs is not really either funk or blues, but sits somewhere between. Responsible for the classic Green Onions.
Jimmy Smith's live album Root Down! is an excellent Hammond organ based album. Combines funk, blues and jazz - so if you're thinking of looking further in those areas this is a great introductory album. Title track was later sampled by the Beastie Boys.
Herbie Hancock's Head Hunters is a definite must-have in any collection. No serious music fan can be without this album! Instrumental jazz-funk and considered one of greatest of the genre. He released several other albums around the same time (early 70s) like Thrust and Man-Child that are worth checking out if you like what you hear.
As I say, I've only touched on a tiny fraction there. There's the infamous James Brown (and the Famous Flames), Billy Cobham (A funky Thide of Sings is an amazing cop-show funk album), Curtis Mayfield (probably the best known of the 'blaxplotation' soul/funk artists), Average White Band...
I'll leave the specific blues recommendations to someone else, but in the meantime a few big names to investigate are Stevie Ray Vaughan, BB King and Albert King.