Psychiatrists don't have special powers when it comes to choosing an SSRI, the process is almost entirely trial and error.

There is nothing wrong with going to a general practitioner, trying Lexapro, and if you don't feel anything on that, trying Wellbutrin, and then on down the line.

If you administer the Beck Depression inventory to gauge your own mood on some consistent basis before and after starting antidepressants you will be introducing far more of the scientific method into the process than most psychiatrists will.

https://en.wikipedia.org/wiki/Beck_Depression_Inventory