Well, to establish an advantage, so that I CAN apply pressure with things like dair.
Okay, sweet (you may want to edit the OP to make this more clear for others who want to contribute).
In order to establish an advantage so you can abuse people in their shields (it sounds like that's your goal), you have to do two things, and you generally have to do them at the same time. You have to: 1] close distance and 2] reduce options for opposing movement/counterattack.
Closing the distance between you and an opponent depends on the matchup and their ability to keep you out or, on the other hand, their ability to close the distance on you and begin applying pressure before you can. In general, Yoshi can close space quickly. He has decent run speed, two big jumps and specials that move him forward while also keeping his momentum. Reducing enemy movement and options are important because you don't want to spend all this time getting in, maybe even taking damage on the way there, only to have someone squirm out or blast you away to have you start all over again. More importantly, reducing options means that the opponent has a higher chance of relying on universal defensive mechanics, which means that your end goal of putting people in shield is that much closer.
The difficult part is putting these two ideas together at once. For example, if you really wanted to, you could just Egg Roll at people to close distance, but because you're not really reducing the enemy's options while moving at them, they can simply jump away and begin running...and this is just the least punishing example. With all this being said, Yoshi has some choices he can make to get the job done.
Keep in mind everything is matchup-dependent and when thinking of establishing an advantage, you have to assume when thinking about these things is that 1] the opponent knows the matchup (always assume this until they've proven otherwise during play) and 2] they are always in a position to run away (i.e., they're not perpetually at the edge of FD and down a stock and fifty percent every time you consider a scenario).
One option (and it's still early, so I don't know how good it is, although it's usually a very safe bet) is to approach on foot and control ground space while inching forward to force the enemy to take risks to get out or to jump away. I mean things like dashing, using down tilt and pivot forward tilts, short hop back air, sliding Egg Toss and Egg Lay. This method is designed to be the safest way in, although it's probably the easiest to get out of. By controlling the ground and spacing properly while inching your way in, opponents can't run past you or roll past you, running up and shielding can be variably unsafe depending on what they shield (with Egg Lay being a good option to stop that if it becomes a pattern) and if your spacing is good then they have a low chance of successfully counterpoking you. Unfortunately, jumping our of this or running further away from this is really easy and is part of the reason Yoshi is so weak to run-away.
Another option is to be more aggressive and reactive by taking a low-flying aerial approach. With this method, you cover more escape options at once, but you are committing more heavily and are more prone to someone attacking you back in order to get you away from them or to set up their own offense. This method involves advancing neutral airs, aerial Egg Toss, falling Egg Lay, full jump back airs and reacting to aerial escapes with double jump. Advancing neutral airs create a moving wall that discourages the enemy's forward movement and ends early enough to perform another action when landing. Full jump back airs discourage single jumps and the duration of the move covers the opponent's airdodge option while having a very good recovery time, making it easy to use and queue up another action after. Falling Egg Lay is here as a detriment to those who run up and shield, expecting to punish a falling aerial attack.
A few notes on aerial Egg Toss. I'm still learning about it--the move is good, but learning to control it can be difficult. Aerial eggs make a great wall because of their new arcs and the recovery on the throw in the air makes reacting to the enemy a little easier than in the past. The problem with aerial eggs is that whether you're throwing them with the intention of covering your descent/approach or throwing them in a more offensive manner with the intention to tag someone, this is still the option that has the most startup and, as such, is the easiest to avoid with movement. Depending on the situation, enemies can simply run under and away from advancing aerial eggs more often than not.
There are a few other ways to handle creating advantage with the intention of forcing your opponent into shield. You can mix and match styles (I'm certainly not suggesting that you stick any one particular method all the time), but you have to keep in mind all the strengths and weaknesses of all these different methods as you switch between them because creating a concrete advantage that opponents have to respect is a core weakness of the character. Making a strong approach in the neutral a core element of your gameplay means working around this very real weakness.