Ok...you have 2 options;
Option 1 - West Bay - This is where all the towers are, and therefore most of the decent bars and restaurants. Depending on your job, you may well be based there as well. Regardless, the rest of Doha is within easy reach. Plus, you have City Center Mall for shopping.
Option 2 - The Pearl - Set away from the main parts of the city, it's the main new residential development and is one of the nicest places to live if you like to walk around and live by the water. That said, they've taken away the alcohol licenses from the restaurants, so a lot of the life has disappeared.
I'd favour West Bay personally. YES, you do pay more money to live in these 2 areas, but then again, you pay more to live in the nice areas in ANY city. Oh, and give up any thoughts of finding somewhere similar to any neighborhoods in major cities! It's not possible...
Good luck!