A car will give you a great deal of flexibility, and while public transport is most probably entirely possible it will mean waiting for buses and trains and cost in terms of taxis to stations so not a good idea if trying to see a lot in a short period of time.
Make sure you rent an automatic, as many cars will have a manual/stick shift gearbox which can be a challenge if you're not familiar with them.
If you're into trains, take a look at a trip aboard the Belmond Royal Scotsman:
www.belmond.com/royal-scotsman-train/
...and don't forget to visit Her Majesty's former floating palace, the Royal Yacht Britannia when in Edinburgh:
http://www.royalyachtbritannia.co.uk/
..and you can also head out to see a Concorde at the Aviation Museum about an hour north of EDI:
http://www.nms.ac.uk/national-museum...de-experience/
Obviously you should look to avoid, or book well in advance for, the Edinburgh Fringe during August:
https://www.edfringe.com/