With white play 1. e4
If black plays 1...e5 play 2. Bc4. From here you've cut out a lot of theory (ie petrovs, philidors, latvian etc) If you want something simple just play a basic formation with f4, d3 etc. Otherwise you can transpose to a wide variety of openings like the Italian (incl evans), King's gambit, Vienna, Scotch gambit etc. There is a lot of variety here. The modern bishop's opening is considered the the second best theoretical 1.e4,e5 opening behind the Ruy. There's a lot of flexibility here plus you're aiming your strongest attacking piece at black's weakest square on move 2.
vs the Sicilian the Alapin gives the most "bang for the buck" meaning the best winning chances for the least amount of study time. This opening (the Alapin) can give you winning chances against the best players in the world yet also gives you a very active position against lesser players.
vs. the French either the exchange or advance is fine and both fit well with the Alapin Sicilian.
vs. the ck play the panov-botvinnik. Similar in structure to the Alapin and is the best chance for an open game.
vs g6 and d6- I would play either the Austrian or the 150 attack.
vs. Alekhine's the exchange variation.
vs everything else- generally play d4 and the main lines. I'll mention the BMD vs the Scandinavian is a strong attacking choice and fits well with the other openings I've mentioned.
As black- vs e4- the best starter opening is 1.e4, d5 2. exd5 Qxd5 3. Nc3, Qa5 (or Qd8). You can get a playable game against almost anybody and the positions are open and tactical with very little to learn. After you feel you want something else play 1...e5 and aim for a Ruy variation other than 3...a6. I would play the classical with Bc5 but there are plenty of other choices. Against everything else look for lines that play d5 quickly.
vs. d4 and everything else play either the Tarrasch or Dutch. Both are complete systems (ie playable against everything), simple to learn and lead to active, tactical positions. The Tarrasch fits well with the IQP positions I mentioned earlier like the Alapin, French exchange and pan-bot. The Dutch fits well with f4 openings like the king's gambit or vienna gambit. Both openings have been played at high levels and are probably good enough to reach GM level with.
Anything I haven't mentioned stick to your basic opening principles of center, development and king safety and you'll be fine.