Linking Animations to Movement

One of things I've been trying to do for a while is achieve a good looking animated character. I can do the basics (start running or jump when a button is pushed) but anything beyond that I cannot begin to figure out. An example is turning to face the direction the player is moving. What I usually do is just slowly turn the player object towards the direction of movement, but this usually looks bland and (at times) choppy. What are some concepts I can use to begin to understand how I can link animations to the players movement? Any input would be much appreciated.

I hope this wasn't too much of a word wall, and if I need to clarify anything please just say so. A good example of what I'm trying to achieve is the animation system from TLOU2. I'm also not looking for code, just ideas or personal solutions.