I bought a mini 4 pro and I want to start flying, recording etc. What do I need to do to make my flying legal in the UK and EU?

For the UK http://register-drones.caa.co.uk - get registered and then label the drone with your operator ID,  although not a requirement do the free test too.  You'll learn a lot.

For EU,  it depends on what the first country you plan to visit is,   but I always plan to visit France,  and maybe change my mind later - https://alphatango.aviation-civile.gouv.fr/login.jsp select English at the bottom right corner.  

Make sure you check the official websites and not some commercial service.
They can have outdated / wrong information.
And also note that every EU country can have its own 'additional' rules and will have NFZ's that go 'on top of' the EASA rules.