This is not Saudi Arabia. Yes, you do have to respect local sensibilities but you will find that life is pretty normal here and that you can dress pretty much anyway you want (as long as is not oo revealing).

There are no parts of Doha that you can't go to. However, as a woman on your own, you may well get stared at in the souk areas on weekends by the thousands of sub continent ex pats that descend on the area on Fridays and Saturdays.

Life is really pretty normal here in most respects. If you haven't been to the Middle East before, the biggest single culture shock will probably be the standard of driving here which is actually pretty good by Middle East standards.