Taxis are plentiful and not very expensive, relatively speaking, in Riyadh. This is the best and easiest way to get around. (See my next post for details about using taxis in Riyadh.)
Women were allowed to take taxis without a male escort (well foreign women anyway).
There was often a language barrier since many cab drivers didn’t speak English. But almost all would nod and say they knew where a place was , usually without really having a clue where to go. So, instead of asking ' do you know where X is?', ask them how to get there. Do thus BEFORE you get inside the cab (and have an idea yourself before you leave home), and if they can’t answer satisfactorily, decline and wave them on.
I recommend getting very familiar with the city and its layout before or as soon as you arrive, since you will likely have to give taxi drivers specific directions about how to get where you want to go. I didn’t find Google maps very helpful, and websites were usually just in Arabic so looking things up online was problematic, so source out good maps in English, and get a good GIS program or apps for your phone before you arrive in Saudi Arabia.
If you take a taxi, be careful where you sit. In private vehicles, usually the women sit in the back, and only men can drive. The windows are often completely shaded (smoked glass) so it isn’t possible to see inside (and see the women inside). Sit in the back seat if you are a woman, since sometimes te cab drivers took it as a sign you wanted to get fresh with them if you sat within touching distance in the front seat.
Even for foreigners, it’s not acceptable for men and women to hang out together in public if not married, so take separate cabs.Some people I know would make plans for a group outing (to meet up at a mall or restaurant), but the men would go in one taxi and the women in another in order to get there, and once there would pair up as if married when sitting in a restaurant.
If going in separate taxis, make sure you have the other driver’s number too, or contact numbers for the other people in the second taxi in your party and vice versa, to help you stay together and avoid one car getting lost and wasting hours trying to meet up again.
Occasionally I would take a taxi to a mall with a male friend, or eat in the ‘family section’ with one male friend, but I always chose someone that it looked like I could be conceivably married to. Some women brought fake wedding rings to wear to ward off unwanted attention and suspicion. Even then it was a big risk. If the authorities had asked to see a marriage certificate and I couldn’t produce one, it was grounds for me to be imprisoned or exiled from the country, or both.
When taking company transportation (vans or mini buses), men and women were allowed to travel together in the same vehicle if the company has permission for this, but even then the foreigners would usually self-segregate. Only married couples would sit together as a matter of course.
There is a pubic bus system, apparently, but I never used it.