I'm talking about where if you press a directional key once you walk over to the adjacent tile and can't do any other actions during the walk. I can get my character to move one tile width, but my trouble is animating the walk and having the character actually walk to the next tile, rather than teleport. The research I've done has pointed me in the direction of interpolation, but I don't really understand how to get it working.
The easiest way is to use linear interpolation (AKA lerp) over time.
Let's say you want you character to move from
posB. You calculate it's current position every frame from those two positions, the current time (incremented by a delta time every frame), and the wanted movement duration, as follows:
float t = currentTime / moveDuration
vector currentPos = posA + (posB - posA) * t
This movement might feel a bit mechanical but that's a good first step, if you want to go further, look into easing functions: