Even as a tourist knowing only English, the best ways to ingratiate yourself with everyone is to at least start with a French greeting—especially when walking into shops, cafes, etc. Americans are far too accustomed to employees greeting
them. If you walk into a place, some quick, intentional eye contact and a friendly sounding "Bonjour!" go a long way. French shop owners/workers in Paris don't make a point of saying, "Welcome to my shop, may I take your order?"
I speak French, but whenever I would get lost in a conversation, I'd simply apologize, explain I'm American, and am having trouble understanding or following (all in French, of course; its worth learning how to say it), then ask if they spoke English or if they could slow down. At that point, they'd express being delighted to speak in English, and I'd keep trying to speak in French. Made for some very fun times with all the people I met.
It's also worth learning how to say a few key phrases in the correct rhythm and tone. French isn't spoken with an English-like cadence.