I would start by trying to work out why it became so mossy in the first place - most commonly it's because the soil is compacted/poorly drained/shaded (or all 3), but it can also be (as with mine) that the soil is light and so poor in nutrients that the grass doesn't grow well in summer, leaving the moss to take over in winter.
If it's the first, improving the drainage should be a priority (usually by forking holes into the turf, and possibly brushing sand into the holes) - if it's the second, fertilising with a high nitrogen fertiliser would obviously help, although any fertiliser would be better than nothing.
As for re-seeding the bare patches, try to find a seed mix which will match the rest of the lawn (difficult, it's probably a lot different from what was originally sown/turfed.)
