I've played this matchup a lot, and my Doubles partner is a Luigi main. I win a significant portion of the matches I play against him, but I would consider us to be about even in skill level since we perform equally well in tournaments.
What to do:
-Recover high
-D-tilt
-B-air
-F-air (not too often)
-N-air
-U-air
-U-smash
Luigi's ground game is pretty good, but D-tilt keeps him at bay. Fortunately, Yoshi has a much better aerial game, so abuse it as much as you can and use platforms. What you want to do is tack on percent early then keep hitting him offstage. Edgeguard with ECEs if he recovers low since he can't really do anything about them. Sometimes you can mix it up by doing a quick DJ ledgegrab as he uses Up + B to get the ledge before him; there's no need to worry about the weak part of his recovery since it will never break your armor. Throwing eggs at him while he's offstage is good too since his poor aerial mobility makes it harder for him to avoid them.
Your N-air, just like his, is a great combo breaker and that applies to this matchup as well. U-air will beat his aerials handily and can be used to keep him in the air. U-smash is great as well for the same reason but is a safer, stronger move that will kill him off the top at around 100%.
When recovering, always recover high unless he's far away and you want to reach the ledge earlier. His aerials break your superarmor pretty early, but oftentimes he won't be able to follow up with another before you can safely touch the stage. Don't be afraid to ECE him away from the ledge.
What not to do:
-Recover low
-Use Eggs from close range
All of Luigi's aerials, especially N-air, eat through your eggs easily. As a result, you won't want to throw eggs too often or too close to him so he can beat them and still punish you. Make sure to force an approach, preferably into the air, with an egg if he's far away and take it from there.
Luigi's D-air breaks Yoshi's armor at around the early 40 or 50 percents, so don't give him that option to gimp you and recover high. If you have no choice other than to recover low, try throwing an egg to cover you.
I know this matchup very well but have trouble explaining it. I wish the videos I had of it were still up since they would better show how I approach it.
Also MrHazuki, I will get to watching your match with Navn. I actually played him last summer when he was in the states with Tomber and beat him in tournament.