The best way to get around Edinburgh is on foot.  Driving can be a nightmare in the city centre due to the one-way system and bus only routes.  Most hotels have free guides and maps of the major attractions.  Be sure to wear comfortable shoes, Edinburgh still has a lot of cobbled streets and this is not the place for high heels. For walking directions, try

The Royal Mile is the spine of Old Town with Edinburgh Castle at the top and The Palace of Holyroodhouse and the new Scottish Parliament building at the bottom.  St Giles Cathedral, John Knox House, the Scotch Whisky Heritage Centre and the Mercat Cross are located on the Royal Mile.  There are several pavement cafes, coffee shops, restaurants and pubs if you are in need of refreshment, and numerous souvenir shops.   Princes Street runs parallel to the Royal Mile, a short walk away.  It is the main street in Edinburgh and separates "The Old Town" and "The New Town".  Princes Street has souvenir shops, but also one of Edinburgh's remaining department stores, Jenners, and several chain stores.  Princes Street Gardens offer a bit of tranquility within the bustling city centre and also offer a stunning view of Edinburgh Castle.  The National Gallery for Scotland, and Royal Scottish Academy are both located on Princes Street.    

If you want to use the hop on, hop off tour buses, they can be picked up on Waverley Bridge (just outside of Edinburgh Waverley railway station). You get a guided tour of the city and can either stay on or get off to visit places in more detail.  There are four variations of these buses, so you should check the routes to find the one that covers what you want to see.  One tour, for instance, goes all the way out past the Royal Botanical Gardens to the Port of Leith where the Royal Yacht Brittania is berthed. 

The Georgian "New Town" is well worth a visit.  A lot of the buildings still retain their original features.

Also see the Tripadvisor page for Public Transportation in Edinburgh - including details for Lothian Buses and Edinburgh Trams.