Every time it rains hard, water just sits there pooling up in the grass area for days. I have no idea how to fix it without doing a huge construction project. Are there simpler solutions or do I have to bring in new soil with a slope?
If the water's sitting there for days, you probably need to actively drain the area, not just aerate it! You can dig shallow trenches (about 30-40 cm deep) that channel the water toward a lower part of your yard or toward the street, and fill them with gravel so they don't look so ugly - that costs way less than regrading the whole thing and it actually works!
I had the same problem in my yard and before bringing in new soil I tried some simpler things that worked pretty well. First is to aerate the grass with a fork or rent a mechanical aerator to break up the soil compaction, then spread coarse sand or compost on the surface to improve permeability, and if that's not enough, dig small drainage trenches with gravel in the lower spots so the water has somewhere to drain toward a drainage point or toward the front of the house.
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