This page goes over general movement in Link's Awakening, and small time savers that you can use all over your run.
Walking through grass is slower than a regular ground tile. You should avoid grass where you can, or jump over it with Roc's Feather. Optimally you want to jump and cut the grass under you so you can reduce the number of times you need to jump.
Roc's Feather has 4 frames of lag on the landing where Link is immobile, meaning it is ever-so-slightly slower than just walking. Avoid jumping where you don't need to.
When an enemy is in your way, you'll need to either move around it or kill it. Hitting it with the lunge from your Pegasus dash is ideal. A slash from your sword immobilizes you for 10 frames. Jumping with Roc's Feather and slashing your sword in the air only immobilizes you on the landing for 4 frames, meaning this is faster than slashing on the ground. Shooting with the bow immobilizes you for 15 frames but you can deal with enemies further away, potentially saving you the time to go kill them with a closer range option, you can combine it with bombs to make a really fast, ranged bomb attack.
Link walks one tile every 20 frames. It takes 47 frames to charge and start a Pegasus dash, and 59 frames to reach 3 tiles, assuming he is moving in a straight line while charging. This means that as a general rule of thumb, it is faster to dash for any straight distance over 3.5 tiles long.
Canceling your pegasus dash with a sword slash takes 10 frames but you can instantly attack in a different direction. Using the Roc's Feather to cancel a dash only has the 4 frames of landing lag from the jump, you can go full speed around tight corners that wouldn’t be possible without jumping, but it is a pretty long distance jump, making it harder to precisely stop where you want. Canceling a dash regularly by holding another direction also takes 10 frames, Link is still moving during those frames but he’s slowing down and he’s only moving in a straight line during those frames.
There are 6 directions for charging a horizontal dash, and only 2 directions to charge a vertical dash. A dash upwards or downwards can only be charged by moving up or down respectively. A left dash can be done by charging while moving left, up-left, or down-left. A right dash can be done by charging while moving right, up-right, or down-right. Keep in mind that you'll dash in the direction that you're moving at the end of the charge, on the 47th frame, you can change direction at any time during the charge, which can be useful when navigating corners.
Important to note, strafing (angling your dash) will maintain the same speed on the axis Link originally started dashing on throughout the dash. When practical, it is most advantageous to angle your dash, rather than charge the dash while moving diagonally, when moving to a destination that is not in a straight cardinal direction relative to Link.
Shallow water in the overworld slows you down, except in the mountains. Shallow water in dungeons doesn’t. However there’s a slope between normal ground and shallow water that always slows you down. Jumping around those things is preferable. You can barely make a jump from shallow water to normal ground if there’s deep water between them and you can’t dash jump to it at all (except in the cave leading to Face Shrine for some reason).