Is it halachically permissible to gamble at a casino?
You can try one of these shiurim.
Firstly, ask your rabbi. It may very well be the kind of place a good Jewish boy (or girl?) just shouldn't be hanging out. Leaving aside the danger of losing a lot of money very quickly. There may even be Chilul Hashem concerns. The Aruch HaShulchan refers to gambling as a blight that has destroyed many lives. I don't think anyone would say he's exaggerating.
But as far as the gambling aspect per se, the Gemara (Sanhedrin 24b) offers two possible reasons why gamblers aren't kosher witnesses. One reason applies only to people who gamble for a living, which isn't your question here. The other reason is that it's sort of "theft", as the people who agree to place their bets don't really think they'll lose. (The Rambam has a slightly different understanding, but that's for another time.)
Thus the Gemara's case seems to be where people are gambling their money against each other. When going to a casino, you're betting against "the house." It would be hard to argue that the casino doesn't really understand the risks and that you're sortof-stealing its money.
(Now if you win, aren't you receiving money "stolen" from a previous player? I suspect you're sufficiently removed from that. ייאוש ושינוי רשות, ואין כאן דינא דמלכותא להיפך)
Those are the issues as far as I can see. But please ask your rabbi.
UPDATE: R' Ovadya Yosef shlit'a is of the opinion that absolutely all gambling whatsoever is prohibited -- but not everyone agrees.
Some types of gambling may even be considered stealing by some. That would be the case if the came falls under the category of "Asmachta" which is not a legal way to obtain money.
Rashi in Sanhedrin 24b defines asmachta as when you put money down on something, based on a fact which is not really a fact (for example, the "fact" that you will win) and you are in control of the situation. This is because Rashi feels that knowing you are in control of the situation means you are so confident you will win, you never really intended to give up the money if you lose. In a card game for instance, you have more control over winning than if you purchased a lottery ticket, or played slots, and if you think you are a star cards player might expect to win so much you never even imagined you would lose the money you bet.
Tosfos takes a different approach and says that it is only asmachta when you have nothing to gain from the agreement. For tosfos, it would seem gambling (where you have a chance of making more money) is not asmachta.
For a halachic answer please ask your rav.
After the whole discussion and back and forth of the sugya the Rivash 432 says that gambling is a disgusting thing and even if someone made a neder not to gamble anymore beis din should not annul his vow.