Attack Pattern: Goombas just move straight forward. They'll march straight into a bottomless pit if you let them. When they bump into a wall, they turn around.
Basic Fighting Strategies: In all of these games, Goombas can be defeated by any attack you can think up. Stomping is the most fun, and highly recommended! However, it may be easier to beat them by using a fireball (all games but SMW2), or eating them (only SMW2).
Advanced Strategies: Since they often appear in groups, stomping several Goombas without landing or knocking a bunch over with a shell is quite easy. This will earn more points than a single attack, and may earn 1-UPs. In particular, you can stomp a lot of Goombas if you've got a tail in SMB3. In fact, Level 2-2 offers unlimited Goombas...if you've got a turbo controller, this offers an opportunity to earn lots of 1-Ups.
Also, Goombas can become handy weapons in SMW2, by eating and then immediately spitting them out.
Finally, you can kill a Goomba that's walking on blocks by hitting the blocks from below - this will knock them down.

Attack Pattern: Once again, Goombas just move straight forward. If you've got a paragoomba, it will fall slowly towards the ground, and upon landing, act like a regular Goomba. Bubble Goombas float in the air until something bursts their bubble.
Basic Fighting Strategies: Unfortunately, they can no longer be squashed in this game, but a spin-stomp will work just fine on a Goomba. Pop a bubble Goomba's bubble by hitting it from any direction, and knock a ParaGoomba out of the air by jumping on it's 'chute. Afterward, kill them the same way.
Advanced Strategies: Eating a Goomba (with Yoshi) or hitting one with a fireball will force it to give up a coin. Also, if you use a regular stomp on a Goomba, it will get stuck on it's back. You can pick it up like a shell, and kick it at other enemies. This will earn you 1000 points instead of the usual 200.

Attack Pattern: Those crazy Goombas just keep moving straight forward.
Basic Fighting Strategies: This game is a bit different. If you use a regular stomp on a Goomba, it will flatten out, but it won't die. It will just start running around like a pancake with feet. I don't think it can hurt you in this state, but the only way to kill it now is with an egg or a "Pound the Ground". Either of those two attacks will kill it right away, and you can also eat it.
Advanced Strategies: Eating a Goomba is the best way to take care of them, as you can earn an egg. Also, keep in mind that you may not be able to eat them if you stomp them first.

Attack Pattern: Mostly they just bite, but Goombas can also spit little spikes at you in this game.
Basic Fighting Strategies: When you first meet Goombas, you'll only have Mario to rely on. If you can do timed hits, you should have no trouble beating each Goomba with one attack. Later, when Mallow joins you, he will also probably be able to beat a Goomba with one timed hit - have him concentrate on them while Mario takes out tougher enemies. By the time you get to the pipe vault, Goombas are chumps. Any character should be able to beat them with a timed hit. Concentrate on the tougher enemies, and then go to Goombas. Mallow, as usual, should fight less powerful enemies than Mario or Geno.
Advanced Strategies: Blocking regular Goomba attacks is pretty easy. Wait until they do a sort of somersault, then press the A button (should be just a moment before they make the biting sound effect). Blocking their spike attacks is harder. Try to press A just before the spike hits you.

Attack Pattern: Goombas are a bit smarter in this game. They wander around in a section of a level, just waiting. If they spot Mario, they'll charge at him.
Basic Fighting Strategies: As with most games, any attack will beat Goombas in SM64. The stomp is the most fun, but it's harder to aim in 3D. Punching or sliding (punch while running) attacks are probably more reliable for beginners.
Advanced Strategies: If you've got a Bob-omb handy, you can use it as a weapon against Goombas. Same thing goes for Koopa shells.

Attack Pattern: As in Super Mario 64, Goombas tend to wander around aimlessly until they see Mario in this game, then they charge him. When fighting Little Goombas, they attack you with a "head bonk".
Basic Fighting Strategies: When you start out, it will take two hits to beat a Little Goomba, using either stomps or the hammer. Try taking Goombas out one at a time using Mario and Goombario together, rather than spreading out your attacks among the enemies. Also, concentrate on Spiked Goombas before regular ones.
Advanced Strategies: After Mario learns to use timed hits, beating Goombas is a lot easier. One timed hit will easily take them out, and you can block their attacks.
To block a Goomba's attack, press A just as its head touches Mario's. This is surprisingly tricky, I find.