Weeds and Moss

Afternoon,

I'm new to this forum and wanted a bit of advice on my lawn. The house is a new build and is about 5 years old.

Previous to this there was 1 house on the plot and the current backgarden was more of a vegetable plot apparently.

We bought the house 1 year ago, so not sure if this problem has been there from the start or not but we've got hundreds of weeds.

I'm currently working my way round the lawn with a one of those speedy weeder things, but its not very speedy at all!!

I cant seem to get rid of them or keep them under control, whats the best thing to do? Also struggle towards the edges with loads of moss, as its sheltered on one side of the lawn from bushes and have 3 quite large over hanging trees.

Any help would be great

Thanks