For seafood you can't beat Doha's old and original Fish Market which is basically on the same road that takes you to QDC.
If you go early in the morning you might even get a look at some of the fish still alive. Very early and you see how the fish are auctioned off to the merchants in the stalls.
If you are seeking the air-conditioned experience then supermarkets are good. The Family Food Center fish market is quite varied. Avoid Cafreefour as it's already been shut down a couple of times for violating health codes.
Can't help with poultry as I don't eat them.