First, unless you make a legal partnership with your partner, you will only be able to stay as a tourist. Once you have this partnership, you can apply for a residence permit. A question of support will almost surely come up in your interview.

Assuming you can get this residence permit, the second phase will start. You will be able to travel and stay longer in other places.

After a few years, you will probably need to decide whether to ask for a permanent residence permit (requirements may vary by country) or apply for European citizenship yourself. Citizenship will come only if you have permanent residence in the country and have not left it for long periods of time. A permanent residence permit is not so strict.

Once you have a residence permit, many things like bank accounts, insurance policy, etc. will become possible. Without this permit, they will be very difficult, if not impossible.

Without a legal partnership with an EU citizen, I think you will have many limitations.