Are any special tests needed, or can a regular doctor see me for my clinical depression?

Differential Dx. There are no specific laboratory tests to diagnose depression. There are tests which should be performed to rule out conditions in the differential diagnosis. Your md should check for hypothyroidism, anemia, hyponatremia, and liver function at a minimum. Screening tools such as the beck depression inventory provide the severity of depression. Your md can treat mild to moderate depression.
Depression. See you pcp and let him decide if this is within his scope of practice. They should be able to treat this.
Need a Specialist? There are generalists or primary care physicians and then there are specialists. You are probably referring to a regular doctor as meaning a generalist. For mild or uncomplicated cases, they are fine. A board certified psychiatrist is a specialist and the best to handle more severe forms of depression as well as other mood disturbances.
No tests. There are no specific diagnostic tests for depression. It is a clinical diagnosis. You should do some medical tests to rule out medical causes of depression like thyroid function test, but your doctor can order those for you.