The game is simple, so is the math behind the game. Every spin has the same chance of producing any of the numbers, regardless of the last 10, 10000 or 10 million spins. Statistically every spin is an independent event, so it doesn’t matter what happened before. So immediately rule out any strategy which guarantees results based on “What is due to come up”. This isn’t saying you shouldn’t bet on your birthday number, or the day you met your partner. If that pleases you, then do so, just understand that even so there is some form of emotional reasoning behind it, you are still just as likely to get any number on any individual spin.
All bets on the American Roulette table have the same house edge of 5.25% except the five number bets which have a house edge of 7.29%. So the first thing to take note of is never play the five number bet. The second thing to take note of is that all other bets are equal, so it is up to you as to how you want to play the table. Some people like to go for the bigger bets, others like the even money bets so they have regular wins. If you have only a few betting units to play with you might choose the lower payouts, or if you have a large bankroll you can go for the thrill of the large payouts.
European Roulette on the other hand has a 2.70% house edge, so immediately you have to realize that they are the preferred tables to play. You can play exactly the same as you would on a American table only taking into account that there is only the one 0 stop. You don’t have to move to Europe either, though it’s a nice place for a gambling holiday, all you have to do is try one of the many online casinos (links can be found on this website).
If you are forced to play on American tables then look out for those that offer the surrender rule, often found in Atlantic City. The surrender rule applies to even money bets on the table, when a 0 or 00 is landed half your wager is returned. On these bets the house edge is only 2.63%, slightly better than European Roulette. Of course the large bets are unavailable to you which may or may not be a problem depending on your preferred betting strategy.