Attack Pattern: Fuzzies attach themselves to wires in the sky, often ones which platforms will travel on. They travel around the wires very quickly, hoping to knock unsuspecting visitors down.
Basic Fighting Strategies: They cannot be beaten with any attack, so just avoid them. They move predictably, so you just have to watch them and time your jumps.
Advanced Strategies: If for some reason you wanted to stomp on a Fuzzy, you could do so using Mario's spin-jump or Yoshi. In either case you will just bounce off of the Fuzzy, without hurting it and without getting hurt yourself. This might be useful if you need to get a little extra distance out of your jump.

Attack Pattern: On the map, Fuzzies bounce around very quickly and are hard to avoid. In battle they latch on to Mario and suck his HP out, refilling their own HP in the process.
Basic Fighting Strategies: Any attack will allow you to get a First Hit on the map, but the hammer and dizzy spin (if you've got the right badge) work best. Any attack will work well in battle. You should mostly stick to regular attacks, they aren't worth wasting FP on.
Advanced Strategies: When you fight the last bunch of Fuzzies, you should have Kooper in your party. If you have 3 FP or more, use his Power Shell attack to reduce all four Fuzzies to 1 HP, then finish them off with Mario and Kooper's regular attacks.
There is one Fuzzy in Koopa Village that you have to sneak up on to get the shell he's stolen. Try to walk towards him very slowly. When Fuzzy attacks, he sucks on Mario twice, pauses for a second, and then one Heart Point will be transferred between Mario and the Fuzzy. Try to press A when he pauses, right before the heart he sucks out appears.