There are 2 cases:

1. one-shot payments (eg. you sell a 1 year membership and don't require to keep the credit card info on file)

2. recurring payments (eg. you sell 1 month membership and require the payment to happen automatically)

First case:

Using Odoo ecommerce is appropriate. You could either have as many product as you have durations (1 month, 6 months, 1 year). When a customer buys one of them on the ecommerce, you will have to manually manage their membership (you will most likely want to use contracts for that) and modify the end date (v7 and v8) or the next invoice date (v9) of their contract. Or you could have 1 product and as many variants as you have durations; the result is the same, it's just displayed differently.

There are examples in the demo data with Silver membership and Gold membership in the ecommerce.

Second case (only in v9):

Using Odoo ecommerce (as in your_odoo_instance.com/shop) is not appropriate because there is no link between the ecommerce and the contract management. You need to use the new contract management of v9 which comes with a refactored backend but also with a new frontend (under My Account under the customer login in the top right corner). The customer can log in, access his account in which he will see his memberships (ie. contracts), his invoices, and his payments. He pays through the frontend of contract management. Contract management can tokenize his credit card information and process payment on a recurring basis. The contract gets updated automatically. For the first payment (ie. enrollment), a quotation can be created and it will generate a contract once it's paid.