Songkhla town has a surprisingly good selection of 'nice' restaurants and coffee shops, especially when compared to neighboring Hat Yai. Although you won't find much that's not Thai, or Thai-ified, you should still be able to find something to suit you. There's nothing too fancy that I found, but there are a few very comfortable cafés and coffee shops.
Seafood shacks on the beach in Songkhla
First mention has to go to the seafood stalls on Samila beach. There are around a dozen shacks in all, set up under the trees behind the wide sandy beach. They all look pretty much the same, with the day's catch set out on ice trays in front, with seating facing the ocean. Since I'm allergic to seafood I didn't even attempt to eat at any of them, so I can't give any specific recommendations.
A simple meal of pork fried rice, with a bunny on top
Elsewhere, there are a number of pubs and cafés along Sadao and Sai Ngam roads, as well as the small streets in between. These include the Crown Bakery, which does a full food menu, as well as E.P's at the Singora Hotel. Speaking of E.P's, they have a branch of their Artisan Bakery at the beach, near the seafood shacks. It's one of the few indoor, air conditioned spaces at the beach, and thus quite popular.
Old school kopi at an old town area coffee shop
There are also some restaurants in the town center area, as well as in some of the old shop-houses of Old Town, where there's another branch of E.P's. For western fast food, there are a few branches of chains like Swensen's and KFC in a small shopping mall that's part of the old town market, about one block south of the National Museum, between the museum and old town.