First go walking in the Pentland Hills just south of Edinburgh - they're easily accessible by bus (to Carlops or Balerno) and the landscape is like quite a lot of upland Scotland, though only half as high as the real mountains.
There are also some good nature sites in the Firth of Forth only a daytrip from Edinburgh by public transport - the bird sanctuary of the Bass Rock (you can go round it in an open boat) or the islands of Inchcolm or the Isle of May for more birds (boat rides from South Queensferry and Anstruther). There are seals all around the Forth, you will almost certainly see some on any boat trip. (You can sometimes see them on Cramond Island, which you can walk to from the shore at low tide and it's only half an hour by bus from the centre of Edinburgh).
That's your first weekend planned out. For the later ones, get a guidebook and talk to people. Representative cities, towns and villages to see that aren't too hard to get to: Glasgow, Stirling (for the castle, far better than Edinburgh's), Culross (preserved 17th century village), Anstruther (mainly for its fisheries museum and the Isle of May), Newtongrange (for the Scottish Mining Museum, an amazing industrial museum centred on what was the largest mine in Scotland), Melrose (for the old town centre and the abbey).