Hello I'm new to this site and sorry my first post is so long

I recently (springtime) moved to a new house and after 20 years living in a top storey flat I now find my self with a fair sized back garden (about 73 ft x 23 ft).
The soil is heavy acidic clay and has been up until this week waterlogged in some areas and at best 'moist' in others it’s south facing and surrounded by a 6 foot high fence on 3 sides, meaning I have good wind shelter and lots of sunshine (when that happens).
The top part nearest the house is taken over by a patio (13 x 13 feet) and 2 huts take up the rest of the 23ft width, the patio is raised and garden 'lawn' is accessed by one step down. The rest of the garden goes down on a slight slope but it's not significant. The patio has a circular concrete thing, not quite in the middle but on the patio, the rest of the patio was covered in fairly chunky stone chippings that had to be removed immediately for health reasons and for want of somewhere else to shift them I moved them all down to the bottom of the garden, I put down a weed fabric membrane before laying them and they take up the entire 23 ft with and 6 foot length, I so wish I hadn't done this as now I have ideas to create a bog garden down there but at the time it was completely waterlogged and I had no idea what to do with it.
Anyway the rest of the garden is 'lawn' except a snake like gravelled path. The previous tenant before the last one was an avid gardener and at some point dug up most of the garden and mixed the soil with shale from one of the 'bings' that occur in my area thanks to the mining of shale for parrafin. I think it's illegal to take this stuff now. I can see the clear outlines of beds that he worked on and will create my beds to follow that because although it's clay it doesn't stick is and well separated by the shale I will of course incorporate topsoil and fertiliser. The problem is the lawn it is prone to water logging in wet weather and is covered in moss weeds and dead grass and I'm not sure what to do with it.
Already I've been forced to choose artificial turf for the patio because there was no other option as it was so uneven underneath and most of my budget has been blown on that. I don't want a picture perfect lawn but it would be nice to walk on it on a sunny day in my bare feet and feel real grass. Thanks for any help you can give me on my lawn and what to do with the gravelled area at the bottom.