The trick to succeeding in Bomberman is understanding effective bomb placement. Bombs are more destructive when they are released in intersections instead of between blocks. When they are placed in intersections, they explode in four directions. When they are placed between blocks, they only explode in two directions. As long as you have one bomb power-up and one flame power-up, you can chain bombs that are placed one intersection apart. This technique is useful for fast clearing of brick walls, and trapping enemies in no-win situations. The more bombs that you can lay simultaneously, and the longer their flames reach, the more likely you are to be successful at destroying an enemy no matter how fast it travels.
There is a downside to this however. When you gain the ability to lay four or more bombs, it can be tempting to lay them down continuously, believing that you will have enough time to escape from the blast. Beyond the fourth bomb, it gets progressively more difficult and dangerous to lay a chained bomb and make it to safety unless you have the Flamepass or Detonator power-ups. If you have the Flamepass power-up, then you have nothing to fear except the touch of an enemy. Otherwise, don't lay more than four bombs at a time, and make sure that when you do, you can reach a safety point before the bombs explode. Even if you have the Detonator, think before you move after you lay each bomb down so that you don't end up trapping yourself behind your own bombs without a Bombpass power-up.
Early stage strategy
In the early stages, the enemies are your biggest threat. You begin the game very weak, with the ability to drop only one bomb at a time, and an explosion range of only one square. This makes it very difficult to hit the enemies with your bombs. Even with one Bomb power-up and one Flame power-up, it's still very difficult to hit a fast moving enemy. The more alleys that an enemy has access to, the better its chances of escape are. Therefore, the best strategy to employ against enemies in early stages is to get as close as you dare, and trap them in closed off alleys. The less room they have to roam, the better. You only need to get as close as your explosion range. In the early stages, it is best to seek out the power-up item in each stage before actively going after enemies to give yourself the best possible advantage. But if an opportunity arises to kill and enemy quickly before the power-up is found, always take it.
Later stage strategy
In the later stages, the biggest threat to your survival... is you. That is, without some of the better power-ups like Flamepass, Bombpass, or even Detonator, you have to be very careful not to set yourself up in a situation where you are caught in your own explosions. By the time you reach the higher levels, you should be very familiar with the bomb explosion timing, so you should know exactly how many bombs you can lay down and still reach a safe place in time. Only expert players should attempt to lay more than four bombs down at a time. If you have the better power-ups, then it becomes easier to stay alive even if you make a mistake. But remember that getting killed means you will revert to a less powerful state and be forced to rely on your reflexes more. As the enemies get more aggressive, you may find yourself in a no win situation, so don't become too reliant on your powers to save you all of the time.
Roughly every fifth stage is a bonus stage (the eighth bonus stage occurs one stage earlier.) During the bonus stage, there are only thirty seconds on the timer, and you cannot be killed. An unlimited number of one particular enemy will appear and you will be able to withstand the explosions from your bombs. You are encouraged to walk around and blow up as many enemies as you can within the time limit. As with ordinary stages, the point value for an enemy rises when they are part of a chain of enemy kills. Because you can't be killed by explosions, if you are fast enough, you can continuously drop bombs and keep a chain explosion going indefinitely, but this won't necessarily help you earn more points unless the enemies are grouped closely together.