When you have your blood drawn for a blood tests the blood that is drawn from you during a blood test then gets sent to a blood testing lab.
In that blood testing lab they will test your blood for just about anything to ensure you're healthy and if they find something in your blood that indicates a health issue such as kidney disease, glucose problems, liver problems, high blood sugar etc.
So there's not really other blood tests but instead the doctor will take your blood through a needle and then send the blood off where the blood will be tested.
You should then hear the results of the blood test within 24 to 48 hours but if not you should contact your doctor.
If your family has a history of things such as diabetes, heart disease etc then the doctor may suggest some other tests as well.
As long as you get at least an annual blood test you should be okay but you can have a blood test done more often if you want too especially if your family is more prone to other health issues including the kidney disease, heart disease, diabetes etc.