Site Collections are the basic security boundaries in SharePoint. You
will need to give them Full Control at the top  level site of each site
collection you create.

do I understand you to say that every time
we create another site, I have to go in and give Full control? And then
for every site that gets created under those, I have to go in again?
The thing is, we only have the top  level portal so far and one team site
and I have to go in and give Full control  for every different admin
page they go to on the top portal and every individual page of the team
site. They keep getting denied access.

I guess I'm panicking here. In my head I'm seeing worst case scenario:
I get hit by a bus and my guys have to reinstall the entire portal from
scratch so someone else becomes an administrator.

There must be somewhere that I could grant  global rights  to two other
admins?

Go to Central Admin > Application Management > Policy for Web Application.

Create a new Policy that gives the group of users Full Control.