Well, I am not the hang out type myself and I am not interested in idle talk as well maa sha Allah.
when I go out it is mainly for my kids , so they don't have to sit around the house all day long :)
There are some organizations here that go visited the sick and help the orphans.
go to the Muslim Expatriate Association' group (Stormy Foundation) and ask about it insha allah

some of the sisters here get together once a week and they have something like a e-book group (Islamic studies ) , if you email me I can get you the email address of the sisters insha allah if you like.

As for learning Arabic I am still looking for that myself.
Somebody said go to the Qatar guest center , I seen the sign by landmark but haven had the time to go actually and find it ... I let you know when I find out more insha allah

for the most part without a ride you will be stuck , it's just like that here:(