What shall we do with our new garden? Pics attached

[eastwest]

So we've bought a house with a fairly mature garden and are gardening novices. Garden is not top priority, but I'm trying to plan and approach improving it sensibly, so looking for any and all ideas.
It faces north, is long and thin and is currently divided into 3 sections by trellises, which we would like to remove since nothing is growing up them right now and we thnk they add to the shade.
The slab paved area near the house is basically in permanent shade. Would like to have containers here and a table/chairs. Considering adding a border with shrubs where the lawn meets the slabs. The concrete slabs are likely to soon be replaced with gravel or something for better drainage.
The back third of the garden is shaded by two large sycamore trees, it was very weedy and we have put material down to cover the weeds until we can figure out what to do with it. This area used to be a vegetable garden and we would like to grow veg again. there is a dilapidated shed which we would one day like to replace/re-site, but again isnt top priority.
There is a mature apple tree about half-way down the garden. Area under it is very shaded, mossy, bare of grass, grows only dandelions and slugs.
General facts about the garden: it's slugalicious and doesn't get much light. Clay soil.
Some things I am thinking of (how practical are these?):

• Make the area under the big sycamore trees more attractive. Currently it's full of small rubble and weeds. Perhaps a rockery? Would that work? Ferns seem happy enough.
• Remove the trellises, turn the back third of the garden back into a veg area (we also have a child who will probs kill all veg with footballs though)
• Landscape the lawn slightly into curves, and put woodchip or something under the apple tree.

We both work so need a garden that is relatively low-maintenance, without wanting to take all the fun out of it!

Sorry about the angle- I found it hard to give a sense of the size of the whole garden with the phone camera. it slopes very slightly towards the house.

One of the sycamore trees at the far end of the garden.

Having some separate "zones" can work well in a long garden, so your ideas sound good. I agree with taking down those trellises, or at least making them a bit smaller. There are nicer ways to divide a garden, such as actual plants / flower beds!

I recommend putting woodchips under the apple tree. Grass (and weeds) just take water away from the tree.

Some types of bulbs and woodland plants should do ok close to the sycamore, but it doesn't actually look like it is that dark in the photo? If it throws a lot of shade on the vegetable area that could be a problem, as most vegetables like a fair amount of sun.

Definitely looks like the garden has potential!

We have a similar shape and aspect, although no large trees to cast shadows so most of the garden is in sun.
In our courtyard bit we have big pots with semi shade loving plants, hydrangeas, some palms, roses etc. On the house, mostly in shade we have a shade tolerant climbing rose. Also in the courtyard we have a batter run, recycled water fountain. ( pic)
Between the courtyard and lawn we have put a strong pergola ( fencing guys put it up) to accommodate the grape vine which was already there but rampant!
We have put a green house and raised veg beds in the sunniest bit, 3/4 of the way down. At the end we have a pond and cottage garden flowers. Can you prune the trees to get more sun?

Thanks, this is really helpful. Child would love a water feature!
Good to know that woodchip could work under the apple tree. It just looks very scrappy at the moment so I definitely want to do something with that area.
I know it doesn't look very shady at the moment, but this is one of the sunniest days we've had all year, and I'm just thinking of what it will be like in the winter... though come to think of it, the leaves will drop off, won't they.
We were told by the neighbours that it used to be a productive vegetable patch till the previous owners got too old to manage it, so am basing thoughts of veg on that... I think once the trellis is down it will make a big difference.
The trees do need pruning - we have already done a bit but there's a limit to how far we can reach.