It is possible to start a membership on the 1st of the month (or whatever date you like) from the admin side:
Here's how to start a new membership on the 1st of the month.
Go into the customer's profile and click the button to "Assign a membership".
Choose the membership from the drop down and set the start date to the 1st of the month.
Click the button to "Create Membership"
Now the customer's membership will start at the beginning of next month & will renew from that point forward according to the way you defined the membership!
Customers can also select the date to start the membership when they make their purchase online: