All of those 30-somethings about today (and most of the 40-somethings) will not be able to recall getting about, on a British Passport, across the continent of Europe when you needed to consider if there were any visas required. Post Brexit, this may change. Brits are lucky in that a British Passport ranks a joint third in the number of countries that may be visited without any need for a visa or some other form of authorisation
(175 countries - ranking just behind Sweden and German).
Before the UK joined the EU, we were still able to get about with relative freedom. You needed to get a permit if you wished to reside in another country, but the process was not unduly difficult. We lived pre-EU in Germany and so, had to apply for and had, an Aufenthaltserlaubnis.
What may happen, is that the EU may see in Brexit, the need to apply the same rules to UK citizens as to many other countries that are not members of the EU and/ or EFTA.
The EU may charge a fee on entry into any EU country or the EU may require you to have a travel authorisation (similar to that of the USA and Australia). This would involve paying a fee on application and the authorisation lasts for the life of the passport or for a set number of years. Worse case scenario is that, like Turkey, you have to pay a set amount in cash on entry to any and each EU country. Each nation within the EU being, of course, a sovereign state.
British Passport holders may well need a Schengen visa. The Schengen area comprises most of the EU along with Switzerland, Norway, Lichtenstein and Iceland (who are part of EFTA). Some countries have applied to join the Schengen area (Romania, Bulgaria and some of the more recent EU members). As the EU puts it "Schengen is for those that embrace the free movement of people within the Community" Schengen visas come as a 60 day and a 90 day product and allow one and/ or multiple entry. Either way, it will become a real irritation for anyone who travels regularly to Europe if they have to keep renewing their Schengen visa (and provide all documentation, etc etc)
UK citizens, in order to obtain a Schengen visa will also need to show travel insurance and that the travel policy must be valid ALL of the Schengen states. If it is a single trip policy, then only one entry would be allowed. Of course, the other thing to note is that a Schengen visa lasts for a maximum 90 days in any 180 days period. There are those countries whose nationals do not require a Schengen visa. It will depend on how "hard" the Brexit becomes, that will dictate if the UK in years to come, becomes part of that list of countries excused from Schengen.
Could be interesting times, ahead.