You're in luck, because not only did I secondary Luigi, but I also main Yoshi. So I can give you double the advice in regards to that match. I would have watched the Shiek one too, but both of those videos were recorded in such a way that they are super laggy, and the audio and video don't synch very well. I saw enough in that match to get a good enough impression of your playstyle, so lets begin.
FIrst.... I will say that watching that Yoshi was painful, especially during the third game. Watching a Yoshi lose like that is not my favorite thing in the world to witness. But I digress. I'll start off with just tips on the Yoshi match up in general. We Yoshi's prefer to be in one of two places: either in your face putting you under pressure, or zoning you out with eggs from a distance. Being in between either of those places is bad for us since if we throw an egg an opponent could be close enough to punish, and if we approach wrong we could get punished for that. Yoshi's rely a lot on their neutral air, which can be punished if done right. Although with the latest patch, spaced f airs and N airs are much safer. Perfect shield and spot dodge and rushing in punishes would work well though. Also, if you fight a Yoshi on Battlefield or Dreamland, be prepared for platform Shinanigans. When the stage is flat, a Yoshi is more likely to egg camp, although on a platform stage, they tend to use that to their advantage. It's easier for them to use their combos with the platforms, then again, it's easier to do some of yours as well. Just keep that in mind. Also watch out for our d air. This guy didn't abuse it too much on you, but that move is one of Yoshi's nastier moves.
Now for the general advice, first of all, you do need to DI a little better. The most obvious example is one the throw. Yoshi's throws are really among the worst in the game, so if you DI to the side, it's really hard for us to do any sort of follow up without really reading what you do next. It is important to mix and match and DI according to the situation, cause yeah, that Yoshi shouldn't have had an easy time doing a follow up after any of his throws. Also.... try to be quicker about your recover, and never charge the green missile in that situation. It's a good idea not to charge it at all actually. It's good to use in short bursts offstage for return, since you're constantly moving. But if you're charging it, then your opponent has an opportunity to punish it. In this case, the Yoshi was able to hit you with eggs constantly. It's better to use missile sparingly so that you can air dodge or react in another way: you have a super tornado jump, so you can afford to go low if you have to. Just be quick and efficient in your recovery moments, although sometimes holding back can work to your advantage in some cases. Also.... don't spam fireballs. That won't work on challenging opponents. Sadly the Yoshi you faced here wasn't at a level that gave you a chance to get more experience within the match up. There's some more advice I could give you, but I don't want to clog things up in this forum chat. If you want to talk about it more with me, give me a PM.