I'll go through my experience of playing all the 1.e4 e5 openings with white (and black).
The Ruy Lopez is very complex. Theory is a mile long, and if you deviate early, you get no advantage with white. With white when you play into it, you will always play into black's prepared variation (Black chooses early deviations, the Open Ruy, Breyer,Zaitsev,Chigorin,Archangel, or whatever he wants.) Black specializes while white has to learn them all, just like in the Sicilian.
The Giouco Piano is a good way to go. White can choose tactical lines (Evans Gambit/Moeller) or positional lines (early d3 or d4 with Bd2). The pieces all go to active squares,and the play is straightforward. In all fairness, it doesn't guarantee white an advantage,but it does allow white to get a very playable game.
The one problem is The Two Knights Defense. 4.Ng5 is very tactical and dangerous to play if unprepared. The Max Lange attack is just a long drawish variation. The best bets for white are 4.d3 (very positional) and 4.d4 exd4 5.e5 (also very positional). I must say that I have had a lot of bad games as black playing against that last variation. It is Roman's recommendation. (He uses the Scotch move order and goes into both the Piano and 5.e5 Two Knights.)
The Scotch Game is very solid. I just have not been able to make the game complicated enough against 4. ... Bc5. 4. ... Nf6 is fun for both sides and lively in play, though.
The King's Gambit is very fun to play. I have had a lot of great games with white. 3.Nf3 is good if black isn't booked to the teeth. There are some very uncomfortable positions for white, if black knows his stuff. 3.Bc4 is the best way to go in the accepted. As black I play, 2. ... Bc5 which equalizes.
The Vienna Game is rock solid and also good. It really isn't aggressive enough for lower rated players. 1.e4 e5 2.Nc3 Nf6 3.Bc4 will almost always be met by the dull Nc6 instead of the fun Nxe4. 3.g3 is a nice solid positional variation that I have used quite a few times. It doesn't offer white a lot, but the games are always interesting.
The Center Game ... I have mixed feelings. I don't do well with it as white, but I suspect that it is very playable. I have defending a lot of nearly lost positions as black against it. If black isn't careful, an opposite side castling/attacking position comes about where white is winning the race. I've had so much trouble as black that I can almost recommend it for white.
The Four Knights is another solid variation. I have only dabbled with it and don't really recommend it (4. ... Nd4!) 4.d4 (instead of Bb5) is a good way to get into the Scotch too, but Bc5 is available there too (although not as good). The Belgrade Gambit is fun, but you spend all your time learning all the exciting gambit lines, when they bash out 5. ... Be7 (leading to a dull game).
There are obviously others, but this is just a basic overview with my experience.