For the JLA:
I don't think anyone would deny
, Batman, Wonder Woman, Flash, Green Lantern, and Aquaman as core members.
Martian Manhunter and Hawkman were members for a long time, but are not now.
Cyborg is a current member.
The original Avengers were Iron Man, Hulk, Thor, Ant-Man/Giant-Man (Hank Pym) and Wasp.
Hulk leaves in the second issue, but is on the team in the movie.
Captain America joins in the fourth issue.
Hawkeye and Black Widow join later, and are in the movie.
The current team in the comics is Captain America, Iron Man, Red Hulk, Hawkeye, Vision, Spider-Woman, and Quake.
As "classic" line-ups, which I think most people identify with the most, I went with:
, Batman, Wonder Woman, Flash, Green Lantern, Aquaman
Thor, Hulk, Captain America, Iron Man, Hank Pym, and Wasp.