It is difficult to know what to recommend, not knowing what you like doing, and I’m not sure I have similar tastes to many, but (presuming you’ve not been before):
Walking up Arthur’s Seat +/- Salisbury Crags is a must. Then down into Duddingston for a pint in ‘The Sheip’s Heid’ maybe?(others would know better than me about that).
Princes Street including the gardens & gallery, George Street, Queen Street inc the Portrait Gallery.
Walk down through the New Town to the Botanical Gardens.
My personal favourite place for coffee is Soderburgh on Middle Meadow Walk. Their Mazarins and their buns are to die for – so have both! Sit outside and watch people walk and cycle past. You could then walk across the Meadows to Edinburgh Cycles and browse some bikes. Then pop to Bruntsfield and Morningside which are very Edinburgh. Some decent charity shops in Morningside too.
Edinburgh buses are frequent and very reliable, a day ticket for unlimited use is very good value.
However, if you have bicycles there are many places that you’d likely not see otherwise. An off road, circular route along the canal, down to Crammond on the sea, along the coast (through Leith) and on to Portobello with fish and chips sitting on the promenade (swimming in the sea optional, although very trendy). Then back along the Innocent Railway cyclepath is highly recommended.
If it’s raining, Vue cinema at Ocean Terminal (near Leith) has very luxurious all reclining seats at £6.99.