v7.3 is quite old by today's standards.
As I recall 7.x should be fairly much the same for your dilemma. But I hope you are aware of a security risk with your version.
Anyway from memory.
Some you have done already.
Create a new group.
GOTO
Control Panel » Site Permissions
Select the new group
Set the permissions for signatures to 0.
As well as what other settings you want to change for a new user.
Save
Then goto:
Control Panel » Registration Settings
On that screen select your new group to be:
Default Group(s) for New Members:
Then if you wish to promote them to a group with more features after x number of posts.
Goto:
Control Panel » Group Management
You will see a box to add a number of posts to elevate them to.
A hint is the default groups do not have that setting set.
