Usually you "counter" an opponent's plays by having your own, stronger, powerplays. Keep your deck's win condition as powerful and consistent as possible, then just put 2-3 "techs" against the cards that give you the most trouble (some useful and generic options are dimension gate, 4-star ladybug, wild tornado, magic drain, but it all depends on the weaknesses of your deck). Don't run counters that are too specific (skull lair, except in a deck built around it), situational (retort) or costly (magic jammer). 

Make sure to have a solution to popular threats (dealing with a 2500-3000 ATK monster on turn 1) and the popular annoying cards that may be used against you (in a beatdown deck, dodging ladybug by running a few level 3 monsters instead of all level 4). Popular is the keyword, you don't need to do anything against say magical arm shield because nobody runs it, even though it can win a game on its own in the right circumstances. 

About individual traps or set monsters, they shouldn't be too hard to "read" after a while, based on what character/monster they play you can have an idea of what it could be. To play around it, just avoid the worst-case game-losing scenarios (attacking into a face-up mirror wall with your only monster, summoning two level 4 into a set ladybug...) and try to make the trade as painless as possible. Against mirror wall for example, you can wait until you have 2 monsters to make sure at least 1 survives to protect you the next turn. Against ladybug, prefer summoning a level 3 to attack set monsters, and prefer attacking with the level 4 first so you still have your other monsters to attack with if it is destroyed. Same reasoning for every annoying card.