The high season is soon going to start, which means the Paris-Nice is not going to be very cheap and you should reserve ASAP. The train might be a little cheaper but the trip is quite long, even with the TGV.
Upon arriving at Nice, you might consider a helicopter transfer to Monte-Carlo versus the bus. The cost is not very high and the flight, although short, is spectacular.
The casino has different sections. There is no particular dress code, except that shorts and such are frowned upon. In order to have access to the Salons Privés (higher stakes), a formal wear is better and yes there are people in the evening who dress to the hilt but it is seldom unless there is a special event. A dark suit and a cocktail dress are most appropriate, that is, if you play on the tables. If you stick to the one arm bandits, jeans are allright. Don't forget to take your passport with you.
If you are going to gamble, I am only allowed to say "merde". No offence intended, it is the usual luck bringing wish.