The best thing I found to discover glitches it to think outside the box. Try everything out, and be thorough and dilligent. Investigate all possibilities, and if you find something that might have potential, research it. How to perform a glitch, where to use use it, etc.
Some glitches are found by sheer luck while playing or experimenting with the game, others are found as a result from observing how the game acts in certain situations. Try to force a certain situation on a game that should not happen. Abuse how certain mechanics combine. (For example,
entering ceilings in Mega Man 2 by abusing the Item 1 detection.)
Don't forget tools such as taking damage, dieing, or pausing the game. Try out what happens if you do that in unusual situations as well. Some games are a lot, lot glitchier than others, but by trying around enough, you're eventually bound to come across something.