When enemies appear on the field, try to strike first. Press R1 or RB to draw your weapon and attack your foe. The three ways you can begin a battle:
Great - You attacked before the enemy sees you. Foe begins battle with 25% less HP.
Good - The enemy saw you but you attacked first. Foe begins battle with 10% less HP.
Poor - The enemy attacked you first. Lightning begins battle with 5% less HP.
Also keep in mind that there is no penalty for a "Poor"-rated attack om Easy mode.
While exploring the world, you can have Lightning dash by holding down the R2 or RT button. Dashing will reduce the time that it takes you to get to your destination, and if you press X or A to jump while dashing, you can leap greater distances. You can also hold down L2/LT or L3 to dash.
Note that dashing will drain the action gauge that appears when you are in the field. When this hits zero, you will no longer be able to dash nor draw your weapon to attack. Keep an eye out for enemies when dashing.