Please note that in all games, fighting Lakitu is a good "Advanced Strategy" for dealing with the eggs. This section just deals with the strategies for fighting the eggs themselves.

Attack Pattern: Lakitu chucks them from his cloud, and they fly towards the ground.
Basic Fighting Strategies: You can fight the eggs with fireballs or invincibility, but they move so fast that it's hardly worth it. Dodge, then wait until they "hatch" to fight.
Advanced Strategies: Run! If you run straight ahead throughout a level containing Lakitu, he'll never hit you with an egg - even if you have to slow down now and again. The only problem is that you can't collect any items, and sometimes other baddies will get in your way whilst you charge blindly ahead.

Attack Pattern: Once again, Lakitu chucks 'em, you dodge 'em. His throwing seems a little more accurate in this game, and he doesn't "broadcast" the throws like he does in other titles.
Basic Fighting Strategies: Once again, fireballs and invincibility (not to mention the tail and hammers) will stop them, but it may be better just to wait for them to land. If you get green, "underripe" eggs, watch out! They don't hatch, but instead they roll around on the ground really fast. Fireballs are your best bet for combatting these, although really daring players might try a tail attack. Just running works well if you have nothing else.
Advanced Strategies: The "charge through the level" strategy I mentioned in SMB still works, although the guys at Nintendo snuck more tricky enemies to dodge along the way in this game. If you can get Lakitu to the end of a level, and have him drop a few eggs, you'll get points (and maybe 1-Ups) for hitting the box while they're on the screen.
SPECIAL TRICK!!! In World 3-4 you'll run into a Lakitu that throws green eggs. A little bit earlier in the level there are two pipes on the ground, with a third one hanging down. Lure Lakitu to this area (it's a good idea to clear out the enemies in the area first). Now, go to the hill to the left, and grab a Koopa shell. Take it back to the pipe section, and stand on the platform in the middle. Kick the shell, and enjoy! As Lakitu throws eggs, the shell will hit them, racking up more and more points, and eventually 1-ups! Meanwhile, you're perfectly safe. This trick is tough to do, but if you need lives, it can fill you up quite nicely. There is a 1-up in the bricks right around the place where Lakitu appears, which you can use to keep you going until you succeed.

Attack Pattern: Same as previous games.
Basic Fighting Strategies: Fireballs and spin attacks will work, but as usual, it may be best to wait for them to land first. Yoshi can eat them, too.
Advanced Strategies: Fireballs will earn you 200 points and a coin (assuming you grab the coin), so they're probably the best attack. Running still works, although it's harder to avoid other enemies than in previous games. As with SMB3, taking Lakitu to the end of the level and letting him dump some enemies before you finish will earn you points and maybe 1-Ups.

Attack Pattern: They fall from Lakitu's cloud.
Basic Fighting Strategies: You can't fight the eggs, only dodge them. If one hits you, you'll go into battle with some Spikies (or Spikesters, depending on which Lakitu is throwing at you).
Advanced Strategies: Like I said, you can dodge them or fight them.

Attack Pattern: Lakitu throws them.
Basic Fighting Strategies: You avoid them. Lakitu doesn't throw very accurately, or very often in this game, so it's not that hard.
Advanced Strategies: There aren't any.