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.
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.
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. Using the Roc's Feather to cancel a dash only has the 4 frames of landing lag from the jump. Canceling a dash regularly by holding another direction also takes 10 frames, but Link is still moving during those frames, making this the fastest method to cancel a dash yet the hardest to control.
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.