You've already received good information on the Forum; i'll offer a few comments based on our visit to Edinburgh last April 28-May2 at the start of a 2 week visit to Scotland.
In Edinburgh we stayed at The Inn Place on Cockburn St. It is a short walk from Princes St (if you take the tram from the airport into the city, disembark just before Waverly Station and walk south. The hotel was super convenient and offered breakfast which we had each morning. We used only public transportation (buses) or walked everywhere we visited. We arrived on a Saturday and after dropping off our luggage, headed for the Grassmarket. We enjoyed a late lunch of "street food" followed by ice cream at a small shop at the southwest corner of the market. Nearby are The Kirk of the Greyfriars and the famous Bobby statue. The adjacent cemetery is interesting to walk through and read the many plaques. The descriptions of the Mortsafes were most interesting. We then went to the national Museum of Scotland, and had dinner there at the Tower Restaurant. This wall all a short walk from the hotel (we're in our 60s and 70s).
The next few days we were there we visited the Castle, St. Giles, had high tea at the Signet Library (this was a fantastic experience--it does require a reservation), visited New Town (had a couple meals there including lunch at the Auld Hundred--great place), visited the Palace of Holyroodhouse, the "new" Parliament, Calton Hill (great views of the City), Princes St. Gardens, and had several leisurely walks up and down the Royal Mile.
You mentioned "in Edinburgh for a weekend", is this literally Saturday and Sunday, or was this a figure of speech? If you're there for more than two days, consider this: You could do a day trip to Stirling/Falkirk from Edinburgh on the train. You'd have an option of either spending the day in Stirling (Castle, Bannockburn venue, Wallace Monument, etc.) or, do as we did, after Bannockburn head for Falkirk (also on the train) and visit the Falkirk Wheel (this was a marvelous visit) and then on to the Kelpies before heading back to Edinburgh). You'll need a taxi to get from the train station in Stirling to the venues noted and also from the train station in Falkirk to the venues, but each taxi trip was brief an inexpensive (less then GBP10.).
Lots to see and do, have fun.
PS While you're in Edinbdurgh, after dinner one evening when it is totally dark, walk up to the Castle, then down the Royal Mile, the night time experience is quite different than that during the day when the Royal Mile is packed with people.