Bert and Ernie are friends? (Laughing out loud). Just kidding.
Now, I only watched a limited number of "Sesame Street" episodes when I was much younger. I just watched it, because of the muppets. But, here is who I know are friends, or who I think are friends, anyway:
Bert and Ernie (Ernie is also friends with Cookie Monster)
Grover, Cookie Monster, and Herry Monster
(I think the 3 are friends, I know Grover and Cookie Monster may very well be. Herry Monster and Prairie Dawn are also friends, and Cookie Monster and Lulu may be friends)
Big Bird and Snaffleupagus (Big Bird may also be friends with Little Bird and Barkley, you know, since I think about it since Little Bird and Alice Snaffleupagus are both girls, they may be friends)
Herry Monster and Prairie Dawn (from the earlier response)
Elmo and Zoe
Oscar and Grungetta (Oscar is also friends with Slimey and obviously the other grouches, except if he is so antisocial point, he does not even like other grouches) Are Oscar and Telly really friends?!? (speaking of which, I never saw Baby Bear, but I really have to agree with the previous response that Telly and him are friends)
The Count and his two girlfriends (Countess von Backwards and Countess von Darling)
Forgetful Jones and Clementine
Rodeo and her horse
Kermit the Frog is friends with every muppet from both series: "Sesame Street" and "The Muppets Show"
Biff and Bruno (sorry if the names are wrong, but you know, the two construction workers)
Muppetmonster
P.S. I will be posting on one of the sections on the site, who are friends on the other muppet show: "The Muppet Show"/"The Muppets Tonight"