Legend of the Galactic Heroes (1988-1997): Space opera in its purest form, lots of character development, political intrigue, and space battles.
Mobile Suit Gundam (franchise 1979-present): The mech anime that gave us mecha in a military setting. The franchise has a lot of different entries starting in the late 70's going on until today. They don't have to be watched in order, most series feature the struggle between those that live in space on colonies and those that live on the earth, various factions use mecha that do battle. Lots of character development, continuity (between the Universal Century shows), and mech battles. I recommend Gundam Zeta, 08th MS Team, the original movie trilogy, Gundam 00, and Gundam Unicorn.
Macross (franchise 1982-present): A mech franchise similar to Gundam, it introduced mechanically correct transforming robots, and love triangles to mecha. It was used as the basis for the Americanized Robotech series (something that polarizes fans to this day). I recommend the original Macross, Macross Plus, and Macross Frontier.
Space Battleship Yamato (franchise 1974-present): Predates Star Wars. There is some great space opera here, features great characters, music, and lots of space battles. There is also a remake that has updated animation that I also recommend.
Armored Trooper VOTOMS (franchise 1983-present): Features mechanical designs from Kunio Okawara of Gundam fame. Very gritty mecha series features what has to be the most badass mech pilot in anime (no whiny protagonist here!). Takes the real robot genre to a new level mechs are a lot smaller, don't transform, and the battles are fierce.
Patlabor (franchise 1989): From the director of 'Ghost in the Shell'. A mech show that uses mechs not in a war but in a law enforcement setting. The films have a more serious tone but the show is a lot more light hearted. Great characters, lots of fun.