I tried to use the snake thing that you can buy from home depot but did not get that much. The water does not go down quickly, it drains so slow as if the pipe is drowning with too much water. What could be the problem here?

That thing at HD maybe too short like a zip it drain cleaner for about 3 http://www.amazon.com/Sierra-Tools-Zip-Drain-Cleaner/dp/B003JVXWHU Great for hair clogs in sink drains many times but your clog is further downstream and this can't reach it. Remove the over flow plate and snake down the over flow with a 1/4" snake at least 12' long. http://www.google.com/products/catalog?hl=en&cp=10&gs_id=12&xhr=t&q=drum+auger&gs_upl=&bav=on.2,or.r_gc.r_pw.,cf.osb&biw=1675&bih=824&wrapid=tljp1325698224301018&um=1&ie=UTF-8&tbm=shop&cid=5440789901510905037&sa=X&ei=t4wET9boNuiIsQKzzcSQCg&sqi=2&ved=0CGMQ8gIwBA#

Follow Bobs advice about the snake and then snake from the overflow. Unscrew and remove the overflow plate,(The one with the drain lever) and pull out the tub stopper. You now have a clear shot at the trap. The way you were going in through the drain it ran you square into a tee that you can't get around. Now feed the cable( 1/4" best or 3/3" with a small tip. You can rent a Ridgid K50 sink cable.) About 2' down you will run into the bend of the trap. Crank and push at the same time to work around it. Your blockage will be found in the lateral pipe going to the stack. You shouldn't have to put out more then 10 or 12 feet. I have found tub and shower clogs to be mostly hair and the worm should auger into it and pull it back. After you snake pour a few pans of boiling water down the drain. This will loosen the grease and flush the mess out into the main. Hope this helps and thank you for rating my reply. TOM