Just want to echo what a few people have said, and also point out more clearly that Edinburgh is a _lot_ smaller than London. A bicycle is a _great_ way to get around town (if you have waterproofs!). If you're in the City Centre, most places you'll want to go are within 1 or at most 2 miles.
If you'll be relying on public transport, it's worth a quick study of the bus routes. Look at where you'll be staying and where you'll be traveling to, and make a choice that'll make sense for you. My experience is that transport is _great_ for getting from outside the city to inside, not so great for getting across town.
So long as you don't have to have bus changes at a time of day when buses run infrequently, buses work just fine. If you can find a place where there's more than one service in the general direction of where you'll be travelling, even better.
Bicycle, walking shoes, well-planned bus trips -- lots of ways to explore this lovely, lovely small city.
:-)