How to fix drainage problems in backyard?

We bought a home in Iowa in January. When we first looked at the home in October, we had been through a drought all summer. We did not notice any standing water and the grass was growing well in all areas of the yard. The yard was not fenced but we have since added a wood fence because of our dog. The soil is sandy parts of the yard and clay in the area but the back door out of the garage. We discovered the problems when we had a snowstorm and then a sudden warm up. It was warm enough to melt some of the snow off the roof but the ground was still frozen. The previous home owner used edging blocks around all the beds that rim the house. He did not angle the downspouts enough so the water is able to run away the the house so we know we have to fix that. There is an area right next to the back door out of the garage that is fairly flat and two downspouts run off into that area. Our neighbors backyard is higher than ours and we've now seen that it drains into this same area, adding to the problem. We don't know whether trenching out and putting drainage tubes under ground would fix the issue or it those would just freeze again next winter? There are a lot of trees in the neighborhood and we noticed that a lot of leaves had blown into our yard. Many of the neighbors are 70+ and didn't get the leaves raked last Fall so we have to consider leaves as an on going issue. It is an ugly mess right now and we are still waiting for a thaw to get to work on this problem.