Alan B. Shepard was the first American in space, flying a sub-orbital Mercury capsule. He developed an inner-ear problem that prevented him from flying for the next several years, then after surgery, he was on Apollo 14, and walked on the moon.
Gus Grissom flew the 2nd suborbital flight of Mercury; the hatch of his capsule opened prematurely, and his capsule sank in the ocean - and he came close to drowning as well. He was the commander of the first Gemini flight, and tragically, was killed in the Apollo 1 fire in 1967.
John Glenn flew the first orbital flight of Mercury; he never flew on Gemini nor Apollo, but flew as a senator on the Space Shuttle.
Scott Carpenter only flew once, on the second orbital flight of Mercury. (Chris Kraft was furious at him for how he handled the flight, and he landed about 400 miles off-target; officially, he sustained an arm injury while dirt biking which prevented him from flying again, but the NASA brass probably wouldn't have let him anyway...
Wally Schirra flew on Mercury-Atlas 8, the 3rd orbital flight, and later on Gemini 6 (He rendezvoused with Gemini 7). He was commander of Apollo 7, but the tension between the crew & ground controllers prevented any of his crew from flying in space again.
Gordon Cooper flew the last orbital flight of Mercury, and later flew in Gemini 5.
Deke Slayton was the only Mercury 7 astronaut to not fly on Mercury at all. He had an heart issue that kept him grounded until his first opportunity to fly - on the Apollo/Soyuz Test Project, in 1975, where he rendezvoused and docked with a Russian spacecraft.