You can try the normal variation (1.e4 c6 2.d4 d5 3.Nc3/Nd2 dxe4 4.Nxe4 Bf5/Nf6/Nd7) in which you will enjoy more space to move your pieces. If you don't pawnstorm blindly the kingside, you will certainly have a very solid position.
You can also try the 4.Bd3 exchange variation (after the moves 1.e4 c6 2.d4 d5 3.exd5 cxd5). You will then attack the black king while black will make a minority attack on the queen's side.
2 Bc4 is, surely, ?! at best.
The 2 knights is mostly solid. Only one line that I know of requires white to be precise. 2 nc3 d5 3 nf3 nf6!? 4 e5 ne4 5 ne2 qb6 to save the knight and then 6 d4 and so on.
There are a few more lines that allow white to be worse right after the first 3-4 opening moves, though.
GingerGM who sort-of advocates this thing, has not played it since many years ago- I guess his memories from the following game aren't terribly pleasant:
I like S.Williams. IMO it's not surprising that he played this dubious gambit, knowing that he has already played 1.f3 followed by 2.Kf2!!!!
It's still no good.
All you have to do is grab the center with d4 and if black's bishop comes out to f5 you should probably exchange for it.
There are a few more lines that allow white to be worse right after the first 3-4 opening moves, though.
GingerGM who sort-of advocates this thing, has not played it since many years ago- I guess his memories from the following game aren't terribly pleasant:
I guess Simon should have played 7 Qxe4, taking with the knight looks losing already. What would you play against the hillbilly?
Sounds like the GM version of the great Bernard Parham.
There are a few more lines that allow white to be worse right after the first 3-4 opening moves, though.
GingerGM who sort-of advocates this thing, has not played it since many years ago- I guess his memories from the following game aren't terribly pleasant:
I guess Simon should have played 7 Qxe4, taking with the knight looks losing already. What would you play against the hillbilly?
I fail to see why Black isn't better already after somne simple plan like 7...Nf6, followed by ...0-0, ...b5, ...a4 etc. Actually that "Hillbilly"b3 bishop is a problem piece in any ...b5 line.
Gawain's 6...a5! effectively refutes the whole variation.
What would I play against the Hillbilly? 1...e5, I guess... 
I think that Nd2 works
This is an entirely different line (KIA), where of course you'd rather not expect to see things like 3...dxe4?! 4.Nxe4?! f5?!
As for the Hillbilly attack, it certainly doesn't look like something I would play normally. 
I don't think that it could be called a solid opening, although it seems that it does lead to some interesting positions, as it sure looks like adventurous opening. But I would probably lose badly more often than not in my 1 hour rapid games. That bishop move at the start is probably worse than bishop move in Bowdler attack, or at least comparable to it. Perhaps 3 ... e6 would be a good response for black, with similar idea when you try to make that bishop worse in Bowdler attack.
As for my question (from white's point of view), based on the answers so far, probably classical variation is the way to go, or perhaps two knights variation.
Lol I thought the OP was looking for something solid and sound.
Play the advance! Get your pieces out and try to crank open the position with f5.
Lots of choices. Geller played games where he simply played main lines, locked down more space, allowed exchanges then won the endgames based on more space , more active king and opponents weak squares [with the pieces that could defend them exchanged]. That has some resemblance to the 'Spanish torture' in some ways. FWIW, I play the Panov against CK and Tarrasch against QG currently because I have limited time for chess and want to have games where I have more activity & more fun.
My question is how to play Caro-Kann as white if you are mostly used to play Ruy Lopez (mostly not playing exchange variation)? I had 2 games in a row with 1. ...c6 and I lost them both, in first one I had plenty of chances but I messed them all up, and the second one I just got trashed.
I am looking for something solid as Ruy Lopez, where I have slight advantage from the opening.