Need help for struggling grass in southwest Pennsylvania

I have a big north-northwest facing corner lot, with seven 30 to 60 year old trees. The yard gets some sun part of the day in most areas; about 1/3 of the lot has poor drainage. We hired a lawn service last year, might as well not have bothered. Have the following issues: 1) lots of moss (more than the last 15 years); 2) have dandelions, crab grass, etc; 3) many tree roots near the surface; 4) the grass struggles.

Limited budget, I would like to simply maintain the grass and keep the weeds from taking over (I don't aspire to the "perfect lawn"). It's mid-April - what should I do first, and then what steps through the season? Thank you for any help! Diane

Personally I would turn it into a shade garden

How do you use the area?

Ditto. There are too many problems to make it into any sort of decent lawn.

1) Lawn grasses are full sun plants. Some varieties will handle a little shade, but none will do well under the sun/shade conditions you are describing.

2) Tree roots prohibit grass from growing in that spot. There is literally no soil where there is a large root. If you draw a circle around each tree where the largest roots are, the ones that are showing, or are only and inch or two under the soil, what have you got left? With that many trees, that old, I'll bet there is so little soil that you will even have trouble planting some individual plants from anything larger than cell pacs.

3) Tree roots compete with the lawn and all other plants for nutrients. At least the other plants might have a chance.

4) Weeds grow where the 'main crop' (in this case lawn) cannot maintain a tight cover. Any exposed soil invites weed growth. I do not see anything you can do to make this area grow a lawn that will be dense enough to keep out the weeds.

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Deluxe version:
1) Kill the weeds. Round up, natural weed killers, hand pull... just make sure you get the roots.
2) Dethatch, if there is enough build up of matter that it needs it.
3) Aerate.
4) Rake the debris.
5) Top dress with organic matter, soil conditioner, compost... blended with sand. Ask what locals use for golf courses. If you are lucky there will be a soils or rock yard that sells a nice blend of sandy-loam top soil and soil conditioner.
This layer needs to be deep enough to hide most of the tree roots, might easily be 1" to 2" deep.
6) Roll this material, so it is all evenly compact, and smooth.
7) Seed with the most shade tolerant grass seed sold in your area, and fertilize.
8) Cover the seed with enough compost or soil conditioner to hide it from the birds. This is not much! Not too deep!
9) Keep it moist until the seed is well sprouted. If any weeds come up use boards to lay on the new grass and walk out to where the weeds are and hand pull them.

Ongoing care will be a problem. The very thin layer of soil over the tree roots is not anywhere near enough to maintain a good lawn. You are hoping to get a decent stand started, and that the roots will grow into the original soil alongside the roots. Even so, as the trees continue to grow the roots will keep on growing, too, and will come up through the soil in just a few years.
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Simpler, longer lasting solution:
Mulch the whole area with the chips that tree companies make when they prune or remove trees. Around here they are delivered free for the asking.
Top dress every few years as the mulch decomposes.
Make the layer of chips several inches thick. I put about 6" thick over anywhere I do not want weeds to grow. Sure makes a difference!

While the 'Simpler, long lasting solution' may be quick and easy, it may not look so good. I would do that, though, then, over the years, add shrubs and whatever you want to the area. If you can find a spot to dig a hole, get a plant to fit that location. Research the size of the plant, tolerance for shade, flower color, deciduous or evergreen and so on to be sure you are making some good choices.

Diane_K is right when she tells you that the road ahead will be a hard slog no matter what method you use, sounds to me the guy's you had last year cut the grass far to short, this must have showed bare soil and add to this, the drainage problems you describe is heaven for moss, weeds just get their seeds to germinate in the bare soil too, so unless you want to spend a fortune adding drainage to the soil you will continue to have the same problem every other year,
Remember moss loves damp /shade so you have given it the best start.
What you you have to decide is, Moss is at least green, so do you want to look at green, the weeds are green so when you cut the grass and Moss the mower will remove most of the flower heads, BUT you will still have the shade and wet soil, Moss wont grow on dry soil.

How we had to drain the soil for a lawn was to dig trenched a spade deep and wide, lay into this trench Clay drainage pipes, these are about the same width as your house water pipes that remove rain water, but they are about a foot long, you lay them end to end and the short lengths allow you to do gentle curves to direct water away from the area you want dried out, they are best laid sort of herring bone pattern as this is the easiest and cheapest way to cover a larger area. you wont end up with a golf tee off area, but you end up with a nice lawn area and each year you can add a layer of special lawn sand that helps keep drainage good and also prevents weeds growing.

I like Diane's suggestion of covering area with wood chips and as you get control of the moss / weeds, begin to make nice shaped beds for planting shrubs, trees ect as the seasons go on, this way you could plant things and have different colours and textures all year round as the seasons change.
Best of luck. WeeNel.

A "lawn" doesn't necessarily need to be turfgrass in order to serve its purpose.
Lots of folks in your situation find a clover lawn to be a decent solution.