There are rights and duties that you have as a British Citizen. You keep these rights wherever you go. These rights say for example how British Police has to treat you, but if you are say in Germany, you are unlikely to meet British Police, so these rights are not very helpful abroad.
You have certain rights as a EU citizen. These rights will be useful to you mostly within the EU. Strange enough, some of these rights that you have as an EU citizen you (a British citizen) have everywhere in the EU, except in Britain! On the other hand, if you visit Germany, you might have EU rights that German citizens don't have.
Next, whatever country you go to, you may stay there long enough to become a resident, or short time to be a visitor. For example, if you visit the USA you have the right not to be robbed or shot. You don't have any right to enter the country (but they let you in because it's good for business), but once you're there you actually have quite a lot of rights, which the USA voluntarily give every person present in the USA. And that applies to people who are there illegally as well. Sure, the police can arrest them (like they arrest people believed to have committed a crime), or they can be removed, but they still have all the basic rights.
So as a visitor you have some rights, enough to say you are not "there at your own risk". As a resident you usually gain even more rights, but also more duties.