If you have ANY opportunity to intern at a major, DO SO!
DO NOT, DO NOT, DO NOT intern at a regional unless that is your only option. You can EASILY get a regional job without an internship or any other connection.
I knew two guys at mesa who had gone to the same school for aviation...both started with me as new hires. One had interned at ACA (IIRC), the other at CAL.
Guess who left mesa after 18 months with 1001 turbine SIC, and is flying international on the 75?
Four years later, guess who is getting ready to upgrade and sit reserve in IAD...for mesa?