First, some terminology. You (probably) did not write down your "private key". If anything, you wrote down the wallet's "seed". The seed is the random list of words that is used by the wallet to generate private keys. The reason for writing it down is that if your wallet is lost or destroyed, it can be recovered using the seed. Note that the seed must be protected because anyone that has it can take all of your bitcoins.
If you are not using the HD mode, then I don't know what you wrote down, but backing up the wallet is important.
So I never need to actually backup or write down the actual private key? It's the seed (12 - 24 words) that needs to be written down?
After reading and researching I've decided to get a hardware key - instead of running offline ubunto or having an offline computer.. a hardware key is cheaper as well as easier it seems.
With an hardware key I get a 24 word seed with the ability to add a 25'th word (passphrase) of choice - this should mean that I can always recover my seed as long as I have the 24 words - and even if someone else gets a hold on the 24 word seed they cant use it - since they need the 25th word (passphrase) which wont be written down anywhere.
I think this should be a quite bulletproof way of doing it..?
Now the only 'issue' is that I'd anyway have to send coins back and forth between the private account and exchange wallets for trading. as I want to be able to have a system where the exchange "locks" the coins until payment is received to ensure safe and secure trades.
You have a good grasp of the basics. I would recommend that you keep experimenting with wallets. However, if you want to trade coins frequently then you may need to keep that in the exchange. But your hodl coins should be placed in safe wallets.