I'm interested in seeing a doctor for a general check-up (female), STD tests (routine), and prescriptions for anti-depressants and ambien. Is it okay to see one doctor for all these things?

Family physicians, internists, and even specialists like OB-GYNs deal with a very broad scope of problems on a regular basis, and this definitely includes basic psychiatric problems.  In my experience, physicians are usually a little more comfortable following or managing a problem that is outside their specialty than making a diagnosis that's outside the scope of their training.  But fortunately for you, diagnosing and initiating treatment for depression and insomnia is well within the scope of practice for any family physician, OB, or internist, as are the components of a well woman checkup.  I think either a family physician, internist, or OB could take care of all your problems, just call ahead and talk to a doctor or nurse in the practice and let them know what you are hoping to get out of the appointment.  They will be able to give you a good idea whether their practice is set up to handle it.  As a side note, sometimes Ambien and an antidepressant can be a redundant combination, there are actually several very inexpensive medication options that would allow you to start taking a single medication to treat both depression and insomnia, talk to your doctor about this.This answer is not a substitute for professional medical advic...

If you have used antidepressants before and you know they work, that's fine. However, if not, I would strongly recommend seeing an actual psychiatrist if this is new. Antidepressants can be very dangerous with certain psychiatric disorders, such as bipolar and borderline, which take a lot of skill to distinguish from unipolar depression, especially in a single visit.

In The Netherlands a family physician could do all that, to see a specialist one would need a referal by the FP. OB-GYN definitively wouldn't touch anti-depressants. Btw a recent Cochrane Collaboration paper Januari 31, 2013 showed no benefits on mortality of all these - in European eyes useless and cost generating - typical American well patient check ups, http://summaries.cochrane.org/CD009009/general-health-checks-for-reducing-illness-and-mortality. The only useful thing is to have your blood pressure taken every few years. Of course we recommend doing the usual validated population wide screening programs such as two yearly mammography for women >50, two yearly Pap smears for women >30, and soon to be implemented yearly fecal occult blood test for men and women >60 and the targeted treatment of chronic diseases such as high blood pressure, diabetes etc. Another way of saving on your health costs!