Do I need to manually move a membership into the renewal process?
That depends on the workflow that you designed for each of the membership types, but to make it simple, the different stages are:
Member receives renewal notification with button to “Accept Renewal” – you can do this instead of the member in the step one of the renewals (in the membership manager) – this is set on the notification settings for each membership type
Optional: if you have set up an approval step, then you must approve each of the accepted renewals
Once approved, the member will receive an email with payment details