Yvonne, first I will tell you there is no easy fix either with chemical or by hand.
The moss needs to be got out as the grass under it will be strangled.
You can hire a scarifier if you do not have one and go over the lawn at high then keep lowering the blades this will remove a lot of the top growth, it will also scatter it about and regrow so it has to be swept off and bagged up, some will go in the compost well mixed.
You will be left with a really patchy lawn if there is any grass left under the moss, now is the time to Aerate, you can hire a machine to do this or using a garden fork start one end and methodically push it into the lawn every six inches or so, good exercise and I put my brain out of gear and listen to the bird song whilst doing that.
You now need to make up a mix of compost washed sand and lawn seed to spread on the area's where the grass has gone and gently rake it in. or sweep it in with a hard brush.
Let that grow which means not cutting the grass for a while then after a cut redo any bare patches with more of the mix.
Come the Autumn you will have a sort of lawn spread a winter feed and weed and leave after the last cut, make sure you water it in if it does not rain at the time.
Spring start to feed and weed about every six weeks or so and use the seed mix if there are still bare patches, two years from now you will have a good green lawn.
You could take off the old grass, (it makes good loam after standing in a stack for a year) and replace with rolls after preparing the soil to lay rolls. Modern grass rolls can come in fine medium and hard wearing so you take your pick.
I used to help look after our bowling green, there is nothing easy about caring for grass and it was very labour intensive and yes even with all the work we put in you would get some moss. I do not want to put you off Yvonne just wish you to know it is hard work.
Moss killer will knock most of it back, I hate chemicals although at times needed, then you will have the black patches to deal with and re-seed probably easier in time and effort it is up to you.
We are always here to help so ask if in doubt.
Frank.