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Ring of Kerry. If you live as close as London and you don't do the Ring of Kerry then you're an idiot. Whole other world.
I am sort of interested in the whole Norman side of Ireland though, the bit people don't really rave about or know about so much.
Of course what people should do is both. It's a small country. You could do Dublin, Wicklow and then from south east to south west in a week and still stop off wherever you fancy.
I did Malin Head to Mizen Head 2 years ago and it was great. Very relaxed and the hostels we stayed at were great. You can cycle for miles without meeting anyone on some roads.
I'm from Kerry in the south west and it is beautiful around there. The Ring of Kerry is a really nice cycle route with some great scenery. If you can do it outside of tourist season though it's better as you get a lot of tour buses around.
They are trying to advertise the Wild Atlantic Way now for cycle touring which would pass through some amazing places.