How best to improve this soil

Hi all!

Had this property a couple of years, I've been focusing on the house and just keeping the garden ticking, but now it's time to make a good plan.

Problem I have:

The soil seems to be heavy clay.

Poor drainage, floods easily when it rains.

Weeds seem to thrive but grass doesn't, soil feels hard and compacted.

I've dug up some of it during gardening and while I'm not an experienced gardener, the soil just looks dead to me, like it has very little nutritional value. This is in both the front and back garden.

Dead patches in high traffic areas, grass is disturbed easily when the soil is waterlogged and just dies.

Seeding it seems to just mean the seeds stay on top and don't germinate.

I've aerated it last year and it didn't make a difference, put seed and feed down with some tough grass product at the right time, no grass grew. I've added a bit of sand (advice at the gardening centre) to see if it helped at all but it hasn't really improved the situation.

I'm at a point where I'm willing and able to do something more meaningful to improve the soil condition in the flower beds on the front and the lawn on the back.

Any guidance or input? Be gentle, I'm new to this!

I'm going enough to put some graft in and I have help when I need it.

With the back garden being waterlogged so much during winter, I considered installing a french drain, but honestly I don't know if that's the right way to go either. I have a gentle downward slope in the back and a children's play area at the bottom of the garden where I could put a soak away under.

Thanks!