The way you take payments for your online training business using Trainerize, is up to you. The most common payment method is through PayPal, however, you can take payment via Stripe, Shopify, Mindbody Online and even Square.
You can setup whatever payment system you wish. Trainerize does not have a payment engine built in. This gives more flexibility to you as the Trainer to use what you're comfortable with.
If you wish to take Paypal payments, see how to insert the PayPal button into your web site.
Currently, PayPal is the only payment option that can be used with the Trainerize Site Designer. Other payment options can be used and embedded into more customizable website design systems like those found at Wix, Squarespace, etc.