We had a 10 and a 12 hour stopover at Doha airport. Lucky we were able to spend it in the first class lounge which had wonderful sleep rooms! I don't believe there is anything much to do at the airport. To me, 3 hours isn't a long time. By the time you land, the bus picks you up and drives you to the terminal as well as when you depart you have to get ready to go on the bus back to the plane that takes a bit of time so you might just have 2 to 2.5 hours. We were going to take a tour of the city in our stopover time. We went through immigration & customs where some of the staff had difficulty communicating in English. We walked outside the airport terminal, in to the heat (OMG it was so hot at about 9 am in Sept & Oct). We didn't feel super confident/comfortable so we went back in to the lounge. So I now have a Qatar stamp in my passport and am a few bucks cheaper and it did waste 45 minutes! lol Sorry not sure what else I can suggest as I didn't really see both terminals.

3/hrs isn't sufficient time to do much of anything. By the time you deplane inbound flight, wait a bit in terminal and then be boarding outbound flight, which is done minimum of 45/min prior departure.

Besides, departing airport, means you have to deal again with entering, queues, security, etc.

Find a nice restaurant, do some shopping, read a book... time will pass very quickly.

I have used the Oryx lounge in Doha airport (economy terminal) for each of my last three trips. It costs $40 and that gives you free drinks, food, showers, computers with internet access or wi-fi for those with their own laptops, large screen tvs and comfortable chairs. My layover is usually around 5 1/2 hours, not sure I'd bother for 3 hours although getting the bus and going through security is very quick in my experience but probably depends on the time of arrival.