It looks like roof line makes up a LOT of your property. So, if you can get a functional gutter system, that's a good start. I think your next step would be to dig some trenches to collect that rain water: Lay some storm drain (perforated 4" pipe) in a pea stone bed, which you could then cover over with soil and sod (you may want to put some landscape cloth over the pea stone to keep the soil from sifting into the pipe) you can direct the water to where you want it to go. You can try to do this yourself, and hope for the best, or involve a landscaping architect. (I don't know the correct term for this) Someone who has experience designing drainage systems, to get it done the first time and the right way. If I thought I could manage it myself, I might try to do it without outside help. But, if I messed it up and did not solve the problem, I might end up spending a lot of time, energy, and money for naught.