The process of counseling varies in several things, like how much you trust the process, the counselor and what the presenting problem is. To best describe it think about a fitness and diet regimen, it takes a while and dedication in order for you to see results. There is also long term and short term therapy.
Deep depression sounds complex, maybe your doctor may be involved to help with medication which also takes at least a month to take effect.
Progress does depend on the person, but mainly don't expect quick results, just a step by step process.
Hope this helps and doesn't overwhelm you. Hang in there and good luck!