If you're in Arizona as your post indicates, you will have far fewer options than those in areas with more flights to Europe. If you can crash with someone in Boston, DC, New York, Philadelphia, Chicago, etc, you will have more options. But if you're thinking of booking two separate tickets to save money (say, Southwest from Phoenix to Boston, then Icelandair from Boston to Europe), make sure you understand all the risks. Here's a good summary: https://www.tripadvisor.com/Travel-g41707-c207311/Newton:Massachusetts:Connecting.On.Separate.Tickets.html
You should also be aware that there is no longer any rhyme or reason to airfares. In March 2017 I was looking at airfares for May 2017. First, according to the "experts," that's far to late to get any deals. Guess what I found - JFK to Zurich for $349, nonstop, with a choice of airlines! Not a typo - three hundred and forty-nine dollars. Then, I looked at JFK to Geneva for the same dates. It was $1200 nonstop; the cheapest one-stop ticket was about $650 with a 12 hour layover in Casablanca (Royal Air Maroc had several routes on sale); and the cheapest one-stop with a more reasonable layover was about $700.
Why was Zurich so cheap? Why was Geneva so much more expensive? Your guess is as good as mine.
Also remember, a bargain airfare is only a bargain if it goes where you want to be. There was a post here from someone who had snagged a bargain to Stockholm (presumably on Norwegian Air). But she had no interest in Stockholm, or Sweden, or even Scandinavia. She wanted to see Italy, and only had 8 days. Guess what - she lost one day getting from Stockholm to Italy, and another day getting from Italy back to Stockholm, and spent who knows how money much doing it - so, not much of a bargain. Of course, if you have more time than money, this can work. But it's important to look at ALL the costs before making a final decision. And on a similar theme, look at smaller airports. If you want to see a swathe of France, look at the cost to fly into Paris and out of Nice or vice versa; it may not be any more than flying roundtrip to Paris, and you'll save time and money not backtracking.