Any suggestions

We are first time home owners, young and absolute novices to the gardening world!
Above is the view of our back garden. Hoping for some advice and suggestions about what we can do. I have taken out all the plants from the front bed - I plan to sow this is into grass for more of a lawn.
Down the side I have removed two ever green troublesome trees and I plan to run a 2 foot wide bed along the length of the side fence and have some nice plants and climbers. 2 foot wide is basically as far as the green plant in the middle of the garden on the right comes out currently.
Out of shot (on the far left), there is a long bed which I plan to grow vegetables in.
Along the back row I will put in a 6 foot fence to close it off - as it stands you can just walk into either neighbours garden and someone's pet keeps pooping at the end of the garden!!
At this stage just hoping for any tips or advice on what you think would be good to do. I want to keep as much lawn as possible ideally.
The lawn is yellow in places and a bit spongy, I probably need to aerate it? There are small patches here and there which need reseeding etc. Its a south facing garden so it sees quite a bit of sun which is good!
Any help or tips is much appreciated.
Posts
Which part of the picture above is not your garden?
The big lollypop tree is where the neighbour's garden begins. The 3 yellow conifers are in our garden and the hedge on the left side is right down the middle of the boundary line, so I would be cutting it back to the boundary line and putting a fence up against it, the 3 conifers have to go as well to allow for closing it off.
Once you have made the boundary secure you may want to plant something to conceal the fence at least partially. If so you can ask on here for further guidance. You say you are new to gardening but want to grow veg. I would think that for this year you would be best starting on that side of things, getting the bed prepared amd maybe getting some starter plants from the garden centre and some herbs to go with them.Your lawn will be a good place to rest and think about the next stage.
Do some research on the ornamental side of gardening, look at books in your local library and online, look around the garden centres to see what kind of plants you like, visit some local gardens. When you have some idea of the kind of style you want, you will be better placed to decide on garden plans and people on here will be able to give more focused advice. Autumn is really the start of the gardener's year, a good time for planting trees, shrubs, bulbs and perennials and you can sow broad bean seeds too. By then you may know a little of what you want and be ready to get off to a flying start next spring!
Thanks
Invest yourself a couple of pounds in a soil testing kit, see what you are working with, do you yourself have any pets, kids? Very difficult to give advice, everyone has different tastes. Every garden has its challenges. I had quarter of an acre South Facing, watering was a nightmare! Looks like you have a lovelly space to work with. Always pleased to see younger folk interested in gardening. They think I am hugely amasing at work, some call me Mrs Goodlife. Course you have to contend with (or work with the British weather, depending where you live in the country.
Thanks Nanny Beach. Does the soil kit give PH readings? Will this then dictate if we need to add certain supplements to the lawn?
We've no pets or kids (yet!), for this summer and into next my soul focus is on the lawn really, I want it really lush and green!
Here's an update of where I am
I gave the lawn a cut last night, then leveled the bare patch on the side and put down some grass seed, then lightly covered the top in soil.
I then used a digging fork and punctured many holes in the lawn in rows, usually about 6 inches apart to try and aerate it. I could only get the fork down around 4 inches in most places, but I did a little circle with the fork when in to open it up more. Seems to be a lot of stone underneath, or else its a shallow bed. It was a lovely night last night, but the dryness really showed me how much moss is still in the lawn!
It rained overnight, so the picture above is from this morning.
When the next dry day comes I may add the Evergreen in to try and kill off the rest of the moss. After that I will rake it out, should I then add more lawn seed and something else in a mix, or is lawn seed on its own enough?
The garden is reasonably flat but along the path on the left side it slopes off, how can I level it? I have heard of people using a sand and soil mix, is this a good idea?
Yes, it will give you a PH, a south facing lawn is unlikely to be waterlogged. You are lucky it rained overnight, best to sow grass seed in the spring, you need to keep it watered, the birds love it! I dont feed or water my lawn, I used to weed it, we are on clay its north facing get a lot of celandines. Now, they die back by May, I dont mind the daisies, we have got a few naturalised bulbs daffs. So I just cut it every 3 days. Am happy with that, I do have a dog (we did have 2) and the grandkids visit quite often, so I do not worry. Sand not soil for under your slabs.