Removal of the pool and play areas will require that you till the soil at least to 6 inches deep to loosen it, as it will be very compacted. If the removals were done 2 or more seasons ago, the moles and weeds and insect activity have helped loosn the soil. You can probably just core-aerate the area to loosen it.
Next get a soil test. You can order MSU's Test Kit online at www.msusoiltest.com
, or purchase one from the MSU Wayne extension office- call first to confirm they have one in stock.It costs $25 and includes the postage to mail the sample to the lab. Results are received about 1-2 weeks after the lab receives your sample.
Kill all the weeds first, using a nonselective herbicide if desired. Glyphosate- but NOT extended control- can be used. Wait the amount of time indicated on the label before disturbing the soil or removing the weed tops or putting any seed down. Always follow all instructions and precautions on the label.
The soil test results will tell you what fertilizer and amendments are needed to grow turf. After the weed killer 'waiting time' is done, Till in the recommended organic matter and any pH adjustments the test results recommends. It may also recommend that you till in some granular fertilizer before seeding. You will want a starter fertilizer, one that contains some phosphorus - the middle number on the bag, such as 5-5-2 for example.
Next, select the type of seed that will will grow best in your growing conditions. Tall fescue is most tolerant of part-shade and requires a little less fertilizer than Kentucky Blue Grass. Fine fescue and turf type perennial rye fall in between the requirements of the first two types. Read some articles here to help you decide:
Now you can proceed to sow the seed of your choice. The most important factor for success is to keep the seed moist- not flooded- until he grass is well established. This could mean watering it 2 -3 times per day until it is sprouted. The sprinklers must be gentle enough not to wash the seed away.
For hydro seed you will probably not have many choices, you will have to take whatever the hydroseeding company has available.
Your soil test will tell you how to fertilize the lawn for the whole season. The mole problem is best addressed by trapping. There are articles about it at the 'home-lawns' link above. There is a good video from MSU Extension here-
If you have questions about your soil test or anything else, please write us again.