I totally understand your dilemma - we also live in a barn and have 15/9/6 year olds. You have to combine the needs of everyone. Our garden can look quite formal, but then an enormous trampoline has landed on the lawn like a UFO! In a way you have three main areas by the look of things - the large garden, the other bit of the L and that shady area where the patio furniture is. Given that you have already chosen to place the play equipment in the big garden, and that is where the grass is for the 'pitch', then that should stay as it is. I would be tempted to plant some robust shrubs/trees round the fence, which is a bit of a bland 'expanse' and it would be nice to have some trees at the end to mask out the roof of the other house, which is quite a heavy presence. I say robust because they will have to take the impact of balls, frisbees etc. but not be prickly, as retrieving the play equipment will be required. I would be tempted to plant some trees with contrasting colours at the end - red leaves, blossom etc. For all round interest. The trees will have the added benefit of deflecting the balls, frisbees etc. back into your garden, reducing the need to ask the neighbours to retrieve them. Plus be sound proofing for them and you. Your garden centre will be able to advise on things that grow quickly.
The next thing I would say is that you should have some idea of a long term plan. Kids are not kids for long, and so any planting round the pitch should also be seen as a backdrop for the sweeping, curving beds you may want to put in when your teenagers stop playing in the garden.
That leaves the other two areas. The shaded patio is useful. We have a lack of shade, which is a problem in really hot weather, so we had to build a gazebo, but that is not a cheap option. The other area should also be 'grown up' and intimate. Personally, I would get rid of that bush in the middle - trees round the edge would be a much better idea, giving you more space. I would put in a water feature because I can't tell you how much pleasure the sound of our fountain gives me when I'm working in the garden in the summer or sitting out having a meal on a warm evening. Again, you need to plant tall stuff against that fence to make it look less boring, and lift the height to give you more privacy, but I wouldn't go any higher than that existing tree, so something with a shorter habit - tall trees would eventually block out light from the house and leave you with two shady areas to sit, when you could do with one sunny one. Obviously, I don't know what time of day the photos were taken, and the path of the sun changes in the summer.
I have just noticed the scaffolding on that picture - gosh that new house looks like it is going to be big - you may need taller trees at the back there, but again, it could be an issue with too much shade.
Definitely work with the idea of being able to sit in the sun and in the shade in mind, so that you can make the most of the seasons. And don't forget that curves add length and depth to a garden and will give the visual impression of distancing you from the other properties. Find out about the speed of growth, height at maturity and density of foliage of any trees you plant for privacy. We had a eucalyptus which we loved because it gave a dappled shade and the leaves seemed to glitter and ripple in the summer, and they are fast growers, but on the downside, we lost it in the terrible freak winter of 2010/11, so they are not wholly hardy and they do grow very tall. But if you don't want a tree which is dense and gives off inpenetrable shade, they are good. Poplars grow quickly, but a lot of people don't like them - I am fond of ours which h