When you add a service, choose its Billing Frequency: one-time, or recurring (a subscription) which can be billed weekly or monthly. The client authorizes the recurring charge at checkout, and their card is then charged automatically on that schedule. You don't need to do anything after setup; payments come to you automatically.
Both sides stay in control. You can manage or cancel any client's recurring payment from your Projects tab or dashboard, and clients can cancel anytime. Clients are notified of upcoming or processed charges by in-app notification and/or email.
If you change a rate card's price, clients already on a recurring plan keep their original price until it's cancelled.
If a charge doesn't go through, see What happens if a recurring payment fails?