Wild birds no longer appear on our tables much because domesticated
birds have long been bred to be meatier, and because wild birds are rather
hard to catch. In Italy and France, though, song birds are greatly prized
by gourmets, probably an echo of the excesses of the Romans.
When did birds first appear? That depends on what you want to call a "bird". For certain Velociraptor was feathered, including large quills anchored directly to the arm bones as modern birds that fly have, but I think few consider velociraptor a bird (wrong branch of Theropoda anyway). Even when birds could be said to fly depends on your definition of "fly". Velociraptor drawing by Arthur Weasley distributed under Creative Commons Attribution v2.5.
Proponents of "Intelligent Design" have argued birds cannot have evolved because only a fully developed wing is useful for flying, so intermediate stages make no sense and wings must have been designed. This has been shown completely false - non-flyable wings are highly useful. A study of immature chuckers (B2) shows they flap their wings so as to create negative lift, forcing them towards the ground. This gives them improved traction allowing them to run up slopes so steep predators cannot follow. A nearly mature chucker can run straight up the trunk of a tree. When the wings are mature the chucker can flap them differently so as to create positive lift and fly.
Another example is the ostrich, which while fast can't outrun a cheetah, yet cheetahs rarely get to dine on ostrich. As the cheetah closes in the ostrich spreads its wings and uses them as airfoils to execute a turn the cheetah simply cannot follow, and soon becomes exhausted. So you can see that velociraptor, while tiny compared to the movie version, was a fearsome predator non-winged creatures would have difficulty escaping.
Currently the earliest bird there is general agreement on is Archaeopteryx, of which 8 fossils have been found to date. This bird lived in the Jurassic, about 150 to 155 million years ago and was about 18 inches long from nose to tail tip. It wasn't entirely "birdlike", having toothed jaws rather than a beak and a long bony tail and it wasn't a strong flyer but it could definitely fly. The photo shows a model reconstructed as best could be determined from the fossil evidence. Photo by Ballista distributed under the GNU Free Documentation License v1.2 or later.
So from all this you can see the transition from "dinosaur" to "bird" is a continuum and occurred over a fairly long period of time, possibly more than once. The exact lineage is still being debated, but in any case our Thanksgiving turkey does give us some idea what dinosaur might have tasted like.Links