Welcome to Doha and congrats on the bus trip! I'm Irish and have been here a year....have yet to work up the courage to take the bus though!
It's refreshing to find someone who isn't afraid to go out alone, etc, but do keep your wits about you. I sometimes walk from my apartment near Al Saad to the likes of Jarir or the Centre and have had to deal with more than staring...twice from a labourer on a building site and thrice from men in dishdash. However, after learning about a recent raping here I sadly won't be venturing out walking alone again. That was the straw that broke the camels back!
Enjoy your time here, go out, take the bus but always be careful and aware of those around you.

Just my opinion and experience, I wish you all the best.