Matt,
To add to the comments already provided, you'll be fine with just a Passport for Western European countries. I doubt very much if they'll ever ask for your Driver's license (however, be sure to take it along as it's good "collateral" if you want to rent Audioguides). Under the terms of the Schengen Agreement, non-EU residents are allowed a maximum of 90 days in the countries which are covered by this.
When you clear Passport control at your arrival airport, they'll stamp the Passport which will indicate date of arrival and airport. When you depart, it will be stamped again to indicate date of departure and airport.
Note that travel to Russia or some of the former Soviet-bloc countries is a bit more difficult and "cumbersome". If you're planning on travelling there, check the Embassy website for each country you want to visit, to determine their entry requirements. Someone else already provided the information for Turkey. I'm not sure of the rules for South America, China or other places. Again, check the Embassy websites.
Your plan to arrive a few days prior to the start of your tour is a good idea! I use the same method, and find that it's nice to have a look around and also get over jet lag before starting the tour. BTW, which tour are you considering?
Happy travels!