I doubt very much that any EU country will demand that British citizens get visas to travel in or out, but yes, it seems likely post Brexit that passport control will apply at ports, airports, international trains and border crossings into Denmark and Sweden. What the rest of the EU that does not have border controls at crossing points will do is anyone's guess.
Presumably international airports will be increasing charges for flights from Britain and a rise in costs is always passed on to the passenger, so yes, travel will become dearer.
What puzzles me is that those who voted for Brexit, like those who voted for Scotland's independence and were so narrowly defeated a couple of years ago, seem to have assumed that prices would not rise if the status quo changed. Of course prices will rise - any excuse is always better than none if you want to raise the price of anything