How Trustworthy are online gaming sites?

I live to play craps, blackjack and slots at local Casinos.
Typically the wife and I will drive an hour or two once or twice a month to visit them.
We have been pretty skeptical about these online ones, as we are about a lot of things online.
Well here the Government just opened up it’s own online Casino, they would have us believe that their online Casino is the only one to be trusted.
How safe and trustworthy are these online Casinos? Will they actually give you the money you win? Do they have fair odds like a real Casino?