Attack Pattern: There are two different varieties. Dark brown ones hop along the ground - they usually do two or three small hops followed by one big one, assisted by their wings. Light brown ones run along the ground, take off into the sky, then fly around for a while, dropping Micro-Goombas. If the Micro-Goombas touch you, you'll have trouble moving around. After a while, the light brown ones swoop back down to the ground.
Basic Fighting Strategies: Stomping them once will cause them to lose their wings, but it can be difficult to do (especially with the light brown ones.) When dealing with the lighter ones, it may be best to try to run past them unless you have hammers, which can probably hit them even in the air.
The dark brown ones are easier to stomp, as long as you time your jump right, and try not to stomp them when they jump really high.
Advanced Strategies: If you get hit by Micro-Goombas, press A quickly to get them off.
It's often possible to perform a double-stomp on Paragoombas. That is, you can land on them right after you stomp their wings off for lots of points. Fireballs work well on the dark ones, hammers are probably good on the light ones. Invincibility is always good, too.

Attack Pattern: Paragoombas (or Winged Goombas, as they're sometimes known in this game), act similarly to the dark brown Paragoombas from Super Mario Bros. 3. They basically hop along the ground, suspending themselves in the air for a short time with their wings. Most of them cannot jump very high, although there may be some which can fly higher than most.
Basic Fighting Strategies: A regular stomp will cost them their wings, and earn you 200 points. It's also more accurate and safe than a spin-stomp. However, a spin-stomp will beat them instantly, albeit for only 100 points. Yoshi can eat them, and they can be destroyed with fireballs as well.
Advanced Strategies: To earn lots of points, stomp their wings off and try to land on them when they hit the ground. You'll get 200 points for the first stomp, and 400 for the second, if you don't touch the ground first. Then pick them up and throw them at other enemies for an additional 1000 points.
The best solutions, however, are probably to either eat them with Yoshi or kill them with a fireball. Both of these are pretty safe and easy to do, and you'll earn a coin for using these techniques as well, which is arguably more useful than points.

Attack Pattern: Very similar to the dark brown Paragoombas in Super Mario Bros. 3, these ones just hop along the ground. The height of their jumps varies, and they can actually stay in the air for quite a while.
Basic Fighting Strategies: The basic stomping technique still works, although it can be trickier with these ones, since their jumping heights vary. As usual, they lose their wings after being stomped once.
Advanced Strategies: Fireballs work well. Dodging them is also a good technique to use in areas where they are very numerous. Who cares if it's yellow-bellied?

Attack Pattern: On the map screen, they fly around and swoop down towards Mario when they spot him. In battle, they attack Mario by swooping down on him.
Basic Fighting Strategies: They're actually quite weak in this game; they only take as many hits as a Goomba, so there's really no difference, except that you can't use the hammer on Paragoombas. Concentrate on one enemy at a time, don't worry about weather they're regular Goombas or Paragoombas.
Advanced Strategies: You can hit them with the hammer on the map screen, with some very sneaky timing. Stomping them is safer, though. To block, press A just as they touch Mario during their swoop.