How to move an object without it completely teleporting somewhere else? Here's a video
I move almost any object in blender up/down, left/right, it immediately breaks and goes past another object. If I use the bottom left menus that show up when you move/scale/rotate, they move fine without breaking. Whenever using the direct arrows, they just break. How do I fix that? It would make the projects I'm doing so much easier if I didn't have to redo their positioning every 20 seconds I decide to slightly move something.
Judging by the video, it looks like you are constantly pressing the middle mouse button while moving the cylinder. Constantly pressing the middle mouse button and moving the mouse switches the axis and moves the object on that axis. This happens again and again which causes the teleportation effect.
You can fix this by:

Releasing the middle mouse button after selecting the required axis.
Instead of using middle mouse button, using X,Y,Z on keyboard to lock the axis.