Offering a free membership as part of a trial is a pretty common way to attract new members. For this step, all you need to do is configure a new membership level or use an existing level if appropriate. More details here on creating a new membership level: Membership Level Setup.
Next step, you will want to create a 'Signup Only' product under your new membership level. This will allow members to signup but not renew. Set the price to zero to make it free. More details here on Managing Membership Products and Prices.
Now that you have everything configured, prospective members can sign up to a free trial. Once they are nearing their expiry,
The simplest method is instruct the member to sign up to a paid product when they are near the end of their trial. They simply need to go to your website, login, and purchase a membership. Since they started a trial they already have a user account. You can also provide instructions in the Renewal Reminder email on how they can purchase a membership. More details here on Managing Automated Membership Emails.
Once they have purchased a paid membership, you, as an admin, can remove their free trial membership (this will stop any further renewal reminders). Just remove them from that membership level. More details here on Removing a member from a membership level.