Go for a prepaid provider and give out as little personal information as necessary. Avoid the major carriers directly because they need a social security number.
That said, mobile phones are inherently not private. No matter what provider you choose, they will be able to track your location using tower triangulation. Even if you give a fake address, it would be pretty easy to identify you if you always have your phone on at home.
I used Thunder for a long time, but I have been giving Arctic a try lately and like it. I will likely give Mlem another shot after the next update. Voyager is another great one, but I have not used it for a while.
Overall, there are a lot of great Lemmy apps and I love that most of them did not get abandoned after the initial wave from the Reddit exodus died down. I think we have more high-quality Lemmy apps than there were for Reddit.