Card payments run through your own Stripe account, on every plan. If you do not have a Stripe account you make one as you connect. Open Account & plan, then Card payments, and press Connect Stripe: Stripe asks for your business and bank details on its own pages and brings you back. In the FullDiary app, the Card payments card on the dashboard and Booking rules connect it too. Only the owner can connect or disconnect Stripe.
Connected and verified are different things. Stripe usually finishes checking you within a day. Until it has, bookings go through without payment and customers pay in the shop, and the page says so. Once it says Connected and ready, customers can pay a deposit or the full price when they book, and unpaid invoices you send carry a Pay button (invoice Pay buttons are on Pro).
Then choose what is asked. Under Shop settings, Booking rules, Cancelling and payment, Payment is your usual: they pay in person, a deposit when booking, or pay in full when booking. A service can follow that or ask for something different in its own editor.
The money goes to your Stripe account and then to your bank on Stripe's schedule. Stripe charges its own card fees; FullDiary takes no commission. Disconnect Stripe stops card payments on bookings and invoices.