As an American who frequently visits Europe, all you need is to enter the EU via a country which has a visa waiver treaty. They check your passport, and grant you an instant 90 day visa. Once you're in an EU country, you're free to cross the borders into other EU countries - there are no more border checks. About the only things an EU passport gets you is the right to stand in the EU line at immigration (which is sometimes faster than the visitor line), and the ability to stay more than 90 days without renewing or applying for a longer-term visa.
After the UK leaves, it'll have to negotiate these waiver treaties, which might take a few years. But afterwards it'll be the same as before for 80% of travelers, except now there will be a border checkpoint at the Chunnel. The other 20% will be traveling on business, so it remains to be seen what'll happen there. I would imagine both sides will be anxious to renegotiate free trade or almost-free trade treaties ASAP to minimize the economic impact in both the UK and EU.
You don't understand how the EU works. As an EU citizen I can decide tomorrow to go and live _and_ work in any of the EU countries; no passport, no visa, no residency permit, no work permit, etc required. I can just get up and go any time I like. You can't do that, and now neither will UK citizens as a result of voting to leave.
As for the trade deal, the Leave lobby have assumed that the EU would beg Britain to come back. But it's already clear that won't happen as the other member states have made it clear a long drawn out divorce is not an option; "you've made your choice, so go now". And, contrary to the Leave campaigns claims, Britain needs the EU more than the EU needs Britain. The only deal likely to be offered is the sort that Norway and Switzerland have. To get free trade they have had to accept free movement of people and make annual contributions to the EU budget; the two very things the Leave campaign were most opposed to.