Cheapest way would be to clear weeds and try for some decent grass. When we were kids our badminton "court" was a section of a lower terrace lawn across which a net was erected. When we got tired of swatting birdies - as often happens with specific recreational set asides, the net came down while the lawn stayed.

Realistically, I would start by cleaning up what you have. This is actually a good time of year to move rocks and brick and anything else the PO dumped back there. I'd move the things like rocks and bricks to an area for stockpiling for now (perhaps behind the shed) and get rid of any trash or woody debris. You may well find uses for the rocks and brick such as creating steps or walls or non-flammable surfaces around a fire pit, so you don't want to get rid of them for now. If you don't have a heavy duty lawn tractor with a dump cart, get one to help you move things around the property and mowing. Learn to ID poison ivy regardless of time of year, with leaves and without. Start removing it by pulling it wearing protective clothing OR plan on spraying glysophate (generic Roundup without all the other herbicides) after it leafs out in spring and repeating every 4 weeks or so with any leaves that reappear. You can still get PI when there aren't leaves, so treat winter PI with the same caution needed for the growing season. Don't burn it - you can inhale the oil from smoke and end up hospitalized - but instead put in plastic bags for disposal. If you are particularly sensitive, you can get it from your pets' fur or handling clothing with urushiol (the rash-causing oil in PI), so keep your pets out of it and wash contaminated clothing multiple times in a separate load from other clothing. If you have deer, mice, voles, squirrels, etc. you will have ticks. You can reduce the number by getting rid of the brushy stuff under the trees, but you won't get rid of them, even with a gravel barrier, because the mammals cross the barrier, bringing ticks. Get in the habit of checking yourself, your dogs, and your kids for ticks every time you come inside (just a quick check) and every night a very careful check at bath or bed time including hair, armpits, ears, and groin. Talk to your vet about tick repellents for your dogs and spray your kids' clothing (not them) with a repellent containing DEET. I can pick up a tick between the house and the veggie garden which is all lawn, and during the spring can find 5-10 ticks a day, but haven't ever (15+ years in our current location) had one embed because I am extremely conscientious about tick checks if I have been outside at all. It takes more than 12 hours for ticks to embed, so if you are checking every night, it won't be an issue. Next spring mark wet areas, perhaps with stakes so you know exactly where you have standing water, where it stays damp for more than a few days, and where you have running water. IME trying to change drainage patterns without heavy equipment is usually unsuccessful (and potentially illegal in a natural wetland) but you may be able to slightly modify them so you don't have standing water and mosquitoes breeding. You can plan to plant long-damp areas with plants that are tolerant of moisture. This winter collect information and photos of rain gardens (particularly for shade), visit web and local resources that have information on wetland plants, and find a bunch of photos of dry creek beds and read up on how to create them so they look tolerably realistic. Based on many years as a teacher at a school with natural drainages through the school property, kids love playing with shallow running water- damming it, floating stick or leaf boats, etc., so don't put your drainage under ground. It usually is more successful anyhow to do a drainage swale or dry stream bed than trying to do buried pipes involves a lot more knowledge since underground the water needs a consistent slope and somewhere lower to go to, and pipes need sizing for your largest storms. Also, burying pipes in that area will be difficult due to roots, rocks, and the difficulty of getting a machine in there. Next year after you have drainage patterns established and have moved out rocks and debris, you can do some leveling, adding soil where needed to fill holes that aren't part of your drainage pattern. If you want to keep healthy trees, don't put much soil over large areas of tree roots, however. Then spend the next year IDing what you have (photos on the Name that Plant forum are a great idea) and deciding what you want to keep such as good climbing trees, plants with good fall color or nest sites or whatever, and finding someone with a good reputation for thinning woodlands and the proper insurance. Only after next growing season will you have enough information to really get to work on getting the area thinned out and starting to replant. You should know that despite the inspiration photo you posted of the neighbor's yard, you probably won't get that result. Most grass doesn't grow well in shade, and you said you wanted to keep more trees. Also you have water issues unlike what appears to be drier in the yard pictured. You may do better to have islands of plantings with mulch over the other areas, though you can try growing grass and then changing over if that doesn't work. Good luck - it looks like this project will keep you busy for quite a while!

Stepping stones and 'pebbles'/gravel is cheaper. I think synthetic and stepping stones is cleaner and lower maintenance. When you put down gravel the weedcloth underneath the gravel decomposes within a few years and you are supposed to remove the gravel and relay weed cloth..otherwise you get weeds in the gravel. (like what's happening to your bocce ball court) Here is one of ours with synthetic. You don't have to cover the whole area with synthetic you can create planting areas with low maintenance plants.

I read the words "can't afford. Please don't skimp on your patio or pool guy. I learned a hard lesson at one home. I went for the cheapest bid on the pool, and I got what I paid for. In spades! He cut into the irrigation for the yard and we had leaks under the concrete patio that we discovered months later. It was an infinity pool on a golf course and it kept flooding the course. He neglected to build a high box so that the pump would be higher than the pool. I ended up hiring someone to fix it all and it cost as much as the highest bid I got. Several of his customers sued him, but he promptly declared bankruptcy.... Also my design drove my fiance at the time, nuts! He is a swimmer, but the pool was too short to do proper laps. Very pretty, but not practical. The Jacuzzi cut into the pool length. Yep agree with @littlebug zone 5 Missouri that you need to be sure heavy machinery can get in there.