1. After you perform a shop you report it, usually online at their website. You submit your receipt or proof(s) of visit, usually as an upload with your report. Within a few days, usually, an editor reviews your work and either accepts your job (and sometimes sends you a grade by email), contacts you for clarification by email or phone, or rejects your shop (and usually notifies by email). You need to watch your email or answering machine to respond ASAP to requests for additional information.
Once the shop is finished with the editor, it is sent on along to the client. Your job is now on its way to being paid, which will usually take 4-6 weeks. When you register with a company you will have agreed to payment by Paypal (and given them your Paypal account number), direct deposit (and given them the routing number and account number) or by check. Generally if you are being paid by Paypal an email will be generated when the money goes into your account.
Very few companies require you to submit a separate invoice, but some will not pay without you submitting an invoice, so watch for that requirement in instructions.
2. You NEVER need to buy any sort of membership. If you are looking at companies or 'services' that charge a fee, back away because you are about to be scammed. There are free lists of legitimate companies to work with. See the first thread in the Mystery Shopping Company Discussion section of this forum.