I would defer to your vet regarding appropriate testing. Often, it is not worth the risk of drawing blood on a small bird, unless there are clear signs of illness.
The vet will probably do a physical exam and ask questions about the bird's diet and general behavior. Nutrition is very important to good health for birds. I would check with the previous owner, if possible, to know how they were feeding the budgie.
If you can bring in a fresh poop sample, they might be able to check it for parasites or other things. Pictures of recent undisturbed poops might also be helpful, since unusual droppings can be diagnostic for certain conditions.
Scaley mites can usually be identified by visual examination. The beak and feet will have a characteristic rough appearance.
Difficulty breathing or panting for breath, fluffed appearance, tail bobbing, wet feathers around the cloaca, and vomiting whole seeds are all signs of illness and may require additional testing to narrow down the cause.
Yeast infection, protozoal infection, bacteria infection, and fatty liver disease are all relatively common ailments that can affect budgies. The symptoms can be similar, but they require different treatments, which is why it is important to get a sick bird checked by a knowledgeable avian vet.
If the bird is otherwise healthy, you will also want to check body condition and weight to determine if the diet needs to be adjusted. It is also very helpful to know your bird's base weight since unexplained weight loss can be an early sign of serious illness, like megabactera infection.
Good luck and I hope the checkup goes well.