- The Avengers #1 - The Coming of the Avengers - Origin/formation of the team.
- The Avengers #2 - The Avengers Battle the Space Phantom - The Hulk leaves the team.
Avengers Vol.1: #262-269 The Once and Future Kang -- #270-277 Under Siege -- #278-285 Assault on Olympus
Avengers Vol.3: #19-22 Ultron Unlimited
It may seem difficult for some readers, as Avengers being the flagship title, is constantly the focus of many events in the Marvel Universe. But most of the titles are fairly well contained and explained for new readers.
Avengers - Origin - Updates the formation of the Avengers to the present era.
Avengers Forever - Kurt Busiek's effort to tie together the entire Avengers' history. May be daunting for some but is a rewarding experience.
Avengers Disassembled - Great jumping on point. Also the start of Bendis' decade long run with the Avengers in various series.
New Avengers - Starts right after the previous event "Disassembled".
These are the essential reads, but Hickman's run on the below are a great starting point as well.
Which culminates in,
Avengers - Time Runs Out