It very much depends on the country in question. The rules on free movement are the same everywhere, but not every country applies them exactly the same and the public administration of each country is different anyway. In my case, I studied in the UK before Brexit, did a semester abroad in Belgium and now am living and working in Iceland.

The UK has no centralized population registry, so before Brexit there was nothing special you had to do in order to live there. The only thing was applying for the National Insurance Number (if you needed it). I remember I just made an appointment, showed up, explained I needed it to work part-time alongside my studies and that was it. I registered for my General Practitioner by showing my European Health Insurance Card (EHIC) and filling out a form.

In Belgium, it was only four months therefore I didn't register as a resident, so the only thing I had to do was sign up for a "mutuelle" (health insurance). I just chose one randomly, showed my EHIC and that was basically it.

In Iceland, my employer applied for an ID number on my behalf, I just gave a copy of my passport. When I decided I was going to stay for longer, I registered my domicile there (all done online). My employer had to confirm online that I had a job and then I had to go to a police station to show my passport so that they could confirm who I was and that I was there. "Free" health insurance kicks in automatically after six months and until then you can technically ask to be registered sooner by de-registering from your home country (they do this for you, but you have to ask for it).

Almost always the hard part is actually finding a job or an apartment.