Is a cold wallet worth it?

I started on Coinbase and now I’m on Kraken. I like them both but don’t want to pay a lot in fees as I invest more and worry about what if the platform goes down or the company does under. If you were to invest 10k plus what would your move be to a more permanent place? Not planning on moving stuff around a lot just keeping it there long term.

I don’t know much about cold wallet mainly the concept. How do you actually purchase and track, if not on an app?