Upon Arrival, Get Some Rest: Air travel these days is stressful and tiring. Your body and mind takes a beating and needs to adjust to time changes, food, the different bed you will be sleeping in, and adjusting to an entirely new environment. I have traveled to Europe well over 150 times. My best piece of advice is to TAKE IT EASY upon arrival. Rest, have a light meal, then wake up the next day on a local schedule. What I mean by that is if you normally eat breakfast at 8 am, lunch at 1 pm, and dinner at 7 pm, do this same at your local destination.
Make as Many Arrangements as Possible BEFORE Arrival: Buy train tickets, tour tickets, make reservations at home. It's much easier and less stressful (probably cheaper too) than picking them up upon arrival. For instance, I buy my train tickets on Captain Train. I reserve my Vatican Tour 2 months ahead of time.
Don't Bring Local Currency With You: If you have it of course, you want to bring it with you. But what I do, is simply take out euros at ATM machines once I get to my destination. No need to pay the expensive fees US banks charge when ordering euros.
Get Health Insurance: You absolutely, positively must have health insurance while traveling in Europe. The best site for getting questions answers is SafeGuardTravel. If you don't need the insurance while traveling that's great. But if you do, it may be the wisest purchase you ever made. It can be very, very expensive to be treated for illness/injury if overseas and if you need emergency medical evacuation, expect to pay tens of thousands of dollars. All those expenses are avoided if you have the correct travel insurance.
Plan an Itinerary: Half the fun of traveling is making up the itinerary. Nothing wastes more time than "hitting the ground" with no plan. Try looking at itineraries offered by travel companies or tour operators for ideas.
Relax for Lunch: The reason you are on vacation is to enjoy yourself. Take a break each day for a long lunch. It will invigorate you and you may wind up doing and seeing more.
Get the Museum/City Pass for Each City: You've seen my blog posts on this. Most cities have a pay one price pass that gives admission, most of the time with no waiting in line or no appointment needed to the best attractions in the city. Paris has the Paris Museum Pass, Rome has the Roma Pass, Oslo has the Oslo Pass.
Buy Airfare About 2-3 Months in Advance: Unless getting your ticket with frequent flyer miles, 2-3 months is the time to purchase an airline ticket. Don't forget to checkout discount airlines like Norwegian or IcelandAir.