I rent a booth in a barbershop. How would I go about putting money aside to pay taxes?

Open an account and put a certain amount in each month.

Yes depending on where you live and your normal tax rate
You look at prior years and see how much you normally have to pay, or estimate your income minus expenses for the business.

Since you also have to deal with the self employment taxes also 15 percent by itself.

And if that is not confusing enough, you may need to be filing quarterly estimated taxes on your business income. But putting 30 percent at least back for taxes would be a good start.