Well for Whtie I like, the London System and the Colle. They won't really give you an advantage out of the opening, they push that to the middlegame. But both of them can be used either positionally or to generate abrupt attacks. There is also the King's Indian Attack which, will work against anything black uses as well, as far as i can recall.
For Black, yes 1g6 is the modern defense. there is also the dutch which will work against both d4 and c4. that will mean there is only e4 to worry about. Then there is the 1.d6 system. I don't know much about it, but Cyrus Lakdawala wrote a book on it called 1. d6 Move by Move and it will work against ANY white reply.
As far as the really wierd stuff like 1.e3, 1.d3, etc...my answer to that is always the same: if the opponent is willing to give you the center, take it, and nurse the advantages it give you. So just respond with e5 or d5 which ever one you think is best.

What is the best one-stop solution system(s)? I have heard some say for White a system starting with 1.g3, and for Black a system starting with 1...g6 (I believe the latter is called the Modern Defense).
There are also others: Hippo, Hedgehog, etc.
Anyone have any recommendations and details for White and for Black?
Sometimes players will throw things at me that I am not expecting, such as, for example, 1.e3, 1.d3, etc.....