For Big Bird - Carroll Spinney, the puppeteer (or Matt Vogel, his understudy, who performs when Carroll is absent) raises his right arm above his head to control Big Bird's beak. The puppeteer then shoves his left hand into Big Bird's left wing. Big Bird's right wing is kept alive by a thin wire that is hidden and attached to the left wing that simply causes the right wing to move up and down, depending on what the left arm does.
To see out of the costume, they either a) remove a feather in the neck so the puppeteer can look out, or b) the puppeteer wears a harness with a small TV attached inside the Bird and he can see what Big Bird's doing from the cameraman's point of view.
For Oscar - There is a hole cut out of the back of Oscar's trash can. Thankfully, Carroll doesn't have to be uncomfortable and force himself to squeeze his body INSIDE the can, he is operating from behind the can and shoving his hand up through the back hole and through the top of the trash can. The barrel that the can sits on is also hollow, so actors can stand inside the can if they need to.
When Oscar walks, they hire a little person to wear Oscar's furry legs and to carry a fake trash can on their body. Back in the old days, little person actor Herve Villachiaze, who played Mr. Roarke's sidekick Tattoo on Fantasy Island, performed Oscar leg duty (as seen when Oscar walks his trash can down the street in the 1971 Season Premiere).
Hope this helps!