There are some great coast walks in East Lothian - as well as the Gullane to N. Berwick stretch that Arthur's Eat mentions above, I also really like
https://www.walkhighlands.co.uk/lothian/baldreds-cradle.shtml (limited parking - best to avoid weekends or go early/late in the day),
https://www.walkhighlands.co.uk/lothian/belhaven-bay.shtml and Barns Ness (the White Sands to Skateraw part of
https://www.walkhighlands.co.uk/lothian/dunbar-dunglass.shtml).
Traprain Law is a short steep walk to a good viewpoint -
https://www.walkhighlands.co.uk/lothian/traprain-law.shtml The walk along the Tyne from Haddington to East Linton is very pleasant -
https://www.walkhighlands.co.uk/lothian/haddington-east-linton.shtml (It is worth visiting the Linn Rocks if you make it as far as East Linton.)
If you fancy something more hilly, there are a couple in the Lammermuirs -
https://www.walkhighlands.co.uk/lothian/spartleton.shtml and
https://www.walkhighlands.co.uk/lothian/lammer-law.shtml (but note that the more convenient parking at East Hopes is currently closed). It's also not too far to drive to the Pentlands, which offer a wide choice of options.