It's gonna vary state to state. In NY and VT, a tester comes out to test your birds for Pullorum, which involves getting a drop of blood. They do this by plucking some feather from under the wing, poking the bird so it bleeds a little, and then collecting that blood in a loop.
You can also opt into AI testing, which a lot of states now also require for shipping, but it isn't something that's just included right away, you have to ask.
Then they also need to look over your premises to make sure the birds are kept in good conditions. They don't expect them to be pristine or anything, but no huge piles of filth or obvious things like dead bodies, etc (yes, my tester told me she has had to fail people for the premise inspection for these reasons before). I usually just clean my coops out the day before the testing and we're good to go.
As for cost, in these two states it's free. In some states it costs money. You could contact the person in charge of the program in your state and ask about that. You can find who that would be
here.
Yes. Getting it done through a vet will probably cost an arm and a leg, though. There's a laboratory, Zoologix, that can do testing for you. I know the poultry respiratory panel, which tests for most of the infectious, incurable respiratory diseases is $95. They offer testing for tons of other things as well. You can see a full list of the testing they offer
here.