I need a vertical elevator that does not use sticky pistons, slime, or ender pearls. I need to go up around 30 blocks with one press of a button or lever. How can this be done?
Minecraft 1.13 (The Underwater Update) added a new feature that makes elevators easier than ever. You don't need pistons, slime, ender pearls, or even redstone!
All you need is water, soul sand, and magma blocks. Both can be acquired from the nether, and magma blocks now spawn naturally in the oceans. If you have a column of water source blocks (not flowing water blocks - they MUST be a water source block) with soul sand beneath, it'll create a jet of bubbles that pushes you and all other entities upward. Similarly, if you have a column of water source blocks with a magma block at the bottom, it pulls you down. You can tell if they're working because there will be bubble animations inside the water. It also doesn't consume your air meter, so you can hang out in the flooded "elevator shaft" and not drown. You can learn more about them on the wiki. It is now possible to create elevators that have a 1x1 footprint on the ground, and go arbitrarily tall, with no moving parts whatsoever.
If you want to get fancy with redstone, you can use pistons to push magma, soul sand, or another block type under the column to switch whether it goes up, down, or nowhere, but that is beyond the scope of your question.
Have you seen this tutorial? It shows you a bunch of good elevators and how to make them. As of the needs you described, I would reccomend looking at the 4th elevator. It uses pistons and redstone, but no sticky pistons or slime.
If you want to look at the showcase and the tutorial and nothing else, you can skip to:
Here is one that uses pistons and redstone, but it does not use any sticky pistons:
You can get even more ideas here.
I hope I have been helpful.