The city is themed around the 3 sisters: Castle --gorgeous, a fabulous view of the city; Arthur's Seat in Holyrood Park (takes about half an hour to climb). The wind at the top will blow you away, as will the view if you climb near sunset. Calton Hill -- capture the grace of the castle accented by Arthur's Seat, Compare the layout of the New town and the Old town. Although the city lights are mesmerizing at night from Calton Hill, stay away from it after dark.
The Palace of Holyrood House is worth going to, particularly because you can compare the modern day splendour to that of the Castle. The Royal Mile is definitely a tourist's must. Stop by the cathedrals and memorials along the way from the Castle to the Palace. If you're the theatre going type, you might check out the Festival theatre a few blocks off the Royal Mile on Nicholson Street. The theatre itself is gorgeous, the shows are popular, the tickets are relatively inexpensive and mostly available.
Quick Tips:
Do not shop on the Royal Mile! It's a tourist trap like there is no other! Princes Street has more variety and reasonable prices. For fancy boutiques, check out George Street, but beware, big names also mean big prices.
The nightlife is vibrant. Families and couples: Grassmarket and Rose Street in New Town with its 30+ pubs and bars. 18-24 year olds: the Cowgate for clubs and bars, although some places on the Grassmarket can be entertaining as well.
Definitely stop in at the Three Sisters Pub on the Cowgate - Edinburgh's most famous bar. Dance fanatics may fall for Espionage, a 5-floor dance club in India buildings (Victoria Street). Jazz lovers can find some talent at the Jazz Cellar #1. There is a different theme for every night, and often, live music. Medina is a more Middle Eastern themed bar -- definitely worth going to (especially for lovers of gin and tonics).
And if you have any extra time, I would visit McEwan Hall, part of Edinburgh University -- it's a GORGEOUS building, and worth your time.
Best Way To Get Around:
If you're a walker, it's the cheapest way to go. Cabs are inexpensive for a tryst here and there, but taking them all over the city can get pricey. Late nights are safe to venture about if you're in a group, but make sure not to get involved in any passing fights. Although I walked around by myself when I was there, I would not recommend doing so -- especially after a drunken night out. Cabs can be hard to find on weekend nights, but many companies will come to pick you up for a very small fee (if anything), so look up a cab company number and call if you can't find a cab -- the phone booths look just like the ones in Austin Powers. The buses are a treat, although you'll be hard pressed to find a double decker. Driving is a nightmare. As a pedestrian, I would watch out when crossing streets -- they don't necessarily drive fast, but they aren't very careful either, although they try to be courteous.