What kind of account do I need to access the REST API?

After your credentials are adjusted so that you can access the API, I would take a look at either the f5-sdk or iCR APIs. Both have the REST API under the hood, but making requests to the f5 and getting back data is pretty easy.

Here's some sample code using the iCR API:

!/usr/bin/env python
from iCR import iCR
import json

f5 = iCR.iCR('my.f5.com', 'userID', 'password')
result = f5.get('your/endpoint')
print json.dumps(result, indent=4, sort_keys=True)   pretty print the json output