Well, it's very simple and easy....

Take your original Qatar ID and a photocopy of the ID to DSM office on the Bank street. Once you enter from the main gate you will see 'Customer Services' counters.. Just approach one of the counter and tell them you want to open and account. He will give you one form to fill your details and to pay QR.100 (you have to go to 1st floor to pay QR100 by CASH).
Once you have paid QR.100, they will give you your account number (Shareholder account number).

After DSM office you have to go to 'THE GROUP' who are the brokers. Their office is just oppt. Toyota Building.

In 'The Group' office you have to deposit some amount in Doha Bank Counter and submit the bank deposit receipt to the 'Customer Service' counter. They will register your 2 telephone numbers (mobile or landlines) from which you can do your shares trading, check market rates and buy or sell you share.

The Group website is here: http://www.thegroup.com.qa

Similarly there are other brokers, you can ask DSM Customer Services to give you list.

That's all :-)

All the best.