You'll find everything you want to know at the Doha Securities Market (now Qatar Exchange) website:

www2.dsm.com.qa/pps/dsm/portal/Pages/DSM_Home

As for brokers, I've been using The Group and am quite happy with their services. They have an office in the DSM building, so you can open an account as soon as you've got your NIN number. Dlala is also supposed to be good, though I haven't used them myself.

www.thegroup.com.qa
www.dlalaholding.com

Happy investing