There is a wealth of info on ground covers on the web. I'm sure your local garden center would also have lots of advice. There are too many variables for one-size-fits-all answers (#1 is: sun or shade? but also type of soil, do you want flowering or not, etc.). I have done this in several areas where grass stubbornly refuses to grow & am happy with the results.
the ground cover itself could help with the drainage issue, as its capacity for drinking up water will exceed that of grass (or weeds!)