had similar issues with our house.
getting moss up by hand is hard work. If you can get your hands on a scarifying machine it will help no end.
Re-sow with a shadow grass variety.
An autumn fertiliser (with anitimoss action) should set you up for spring. Some have stuff to alter the pH of the ground which will also favour grass over weeds.
I find that dandilions are best taken out individually by hand. make sure you use a tool to get as much of the root as possible.
Clover seems to be best killed by chemical means.
If removing big patches of moss leaves the place really bald, you can always selectively patch with bits of turf. In the next years it should even out ok.
The only remaining problem I have is our giant dog killing patches with his wee.