Where do you begin?
Step: 1: If you don't have a passport get a passport. Or if your passport is within a year of expiring renew your passport. On a side note related to paperwork only 2 European countries require Americans to have visas: Belarus and Russia.
Step 2: (Optional but recommended, in particular if cost is a factor): Do some research on Europe and European cities and how much each one will cost. I personally like Price of Travel which lists price comparisons for 56 cities ( http://www.priceoftravel.com/2427/europe-3-star-traveler-index/). But compare those prices with the costs of plane tickets. Also do a little research on what each city has to offer. Finally pick your city.
Step 3 (or 4): Buy plane tickets. It is really no harder to buy tickets for Europe than it is for the United States.
Step 4 (or 3): Book a hotel. Do some research before you book any hotel. Tripadvisor has good info, but if you book through Tripadvisor you will likely pay more than if you book the hotel directly. For a few cities in Europe where at certain times hotel space can be limited (for example Oslo) book the hotel first to make sure you have a hotel.
Step 5: Gather information. I recommend investing in guide books you pay for rather than just using information on the internet. While I've found wikitravel pages can be helpful, almost every wikitravel page had information that I found was incorrect when at my destination. When it comes to travel info you get what you pay for. But when you are paying for travel books beware of books that are to short as these books may have been written by someone who just copied info from the internet.
What books do I recommend? Well, I'm biased so of course I like my own Kaufman Green Guide series (Links included at the bottom of this page). But I also like Rick Steve's books and have found there are many good inexpensive independent e-books.
What information should you gather? It's easy to get caught up in the exciting part of travel and forget some of the most practical. How will you get from the airport to your hotel (and perhaps what are the cheapest ways to do so)? What is the general layout of the city? Are certain attractions closed on certain days? Are there tickets you should buy in advance (for example it works well to buy tickets in advance to the Parliament in Budapest).
Happy Travels!