For fun you can go to the souq, sit at one of the restaurants, smoke sheesha if you like it. You can go to shopping malls. You can drive around the desert and spot some camels. You can go to the souq and find different things. If you like the beach, you can drive to the beach and spend some time there, you can even have a barbeque if you like it.There are also lots of nice restaurants in Doha, mostly in hotels. A lot of the hotels also have pubs that you can go to and have a drink.
I do not know if there are English libraries but there are some bookshops like Virgin and Jarir.
There are tons of tailors here.
There are tons of shopping malls and supermarkets.
You can meet people at all those places above and you can also become an active member of QL and join the social group, they organize nice outings.
Good luck!