For saving money, get all day tickets (buy on board) for the local buses. Lets you hop on and hop off as you like.
The National Gallery - also free - on the Mound has a nice collection. The National Portrait Gallery on Queen St has recently been refurbished, and is also pretty cool now.
I like St Giles Cathedral, if only for the Chapel of the Order of the Thistle - it's a very small fee to get it, but its gorgeous.
Go out on the Maid of the Forth - a boat trip out in the Firth of Forth.
http://www.maidoftheforth.co.uk
Or if you are coming in Spring, get the train to North Berwick, and go out to see the gannet colony on the Bass Rock.
Edinburgh Zoo has pandas and a great penguin colony.
in short, Edinburgh has a shed load of different kinds of tourist attractions... I guess if you really don't know what to see/do, it might be better investing in a guide for the city and starting from there? Anything we say/suggest will be just our random suggestions out of a plethora. What is it you like to see/do? And when are you coming?