What do you call the strips on shoes that are often used instead of laces?

Solution 1:

It is called Velcro strips:

the brand name of a type of fastening tape consisting of opposing pieces of fabric, one piece with a dense arrangement of tiny nylon hooks and the other with a dense nylon pile, that interlock when pressed together, used as a closure on garments, luggage, etc., in place of buttons, zippers, and the like.

(Dictionary.com)

Usage examples from Lexico.com

I can't manage things like zips, so they took the zips out and put Velcro in instead.

You may find yourself going out and buying shoes that have Velcro fastenings instead of laces as the child cannot grasp how to tie their own laces.

Solution 2:

To answer the second part of your question, I would use do up informally instead of lace, as

Go ahead; I just need to do up my shoes.

Or, more formally, fasten, as

The girl fastened her shoes before joining the game.