Star Control (1) (1990, Accolade) centered around arcade-style spaceship combat, but included a strategic "campaign game" where opposing sides explored a small galaxy and could build mines, colonies, etc.
Star Control III (1996, Accolade) was not good. In addition to the arcade combat, it had a cheesy adventure game with bad dialogue and many bad design elements. The combat wasn't as bad as the adventure game, but had some balance issues. -MikeBuchanan
Star Control II (1992, Accolade) was the best. Like its precursor, it has arcade-style spaceship combat, now augmented by a 4xish system of exploration, resource harvesting, and ship upgrading, all contained within an epic adventure game. Your character was born on a splinter colony of humans resulting from a failed mission. On that planet, ancient technology from a mysterious race known as the Precursors was discovered and coaxed into building a flagship. You have a rapport with the technology and so are chosen as the pilot. Upon returning to Earth, you find it enclosed in a slave shield by the evil Ur-Quan, with an orbiting starbase your only contact. You take it upon yourself to find allies against the Ur-Quan and free the races they have enslaved. Your task takes you all over a star quadrant, exploring, mining, fighting, and negotiating. -MikeBuchanan
WillShipley agrees (mostly) with the above and is willing to take anyone, anywhere, any time, in StarControlTwo (or III) battle mode.
AndrewFarmer agrees with the above, and would like to point out that there's a Free version, called TheUrQuanMasters?, at http://sc2.sourceforge.net/. Frungy! Frungy! Frungy!