-Nice spike.
-Like Peffico said, avoid using egg roll too much; it's punishable. You especially want to avoid breaking out of egg roll right next to your opponent (0:50).
-When the opponent is at low percentages, you shouldn't use dash attack because your opponent can counterattack before your attack animation ends (0:14).
-I'd also advise against using down-B to get back down to the ground (even though you did it successfully at 1:34), unless you're trying to grab the edge. That said, the aerial down-B at 1:31 was fine because the Lucas was still in his PK thunder lag.
-At 2:20, I thought you had many better options than shielding, especially considering how high Lucas jumped. You usually want to avoid using Yoshi's shield unless you have no other choice (if the opponent is using a long-lasting move, like Marth's side-b and Zelda's upsmash). His oos options are generally terrible (slow roll, 16-frame shield drop lag, and a slow (17-frame?) shield grab). Instead of shielding there, you could have spotdodged, run away, egg-toss slide (which falls under running away), pivot grabbed, or run in and upsmashed.
-Eggs are a nice projectile. Use them when you have some distance between you and your enemy and when he's getting back to the stage.
-Pivot grabs are a good defensive option; back away from their attack, while grabbing them, which also leads into chaingrabs and other grab-release fun (depending on the character).
I second Peffico's offering of good luck.