Quote from: camaxide on December 08, 2017, 04:31:03 PM

So I've downloaded and run the Bitcoin Core client and synced it, set up a passphrase and backed up my empty wallet and written down my privatekey.

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.

Quote from: camaxide on December 08, 2017, 04:31:03 PM

Do I need to anything more before buying bitcoins?
I've looked at Localbitcoins for buying bitcoins - but I know they have a fee. As soon as I make a user at their site I assume I get a new wallet and new privatekey through them? And if so, can I then transfer those bitcoins to the wallet I have in Bitcoin Core without having to pay a fee over again? after all those are both my own wallets.

Localbitcoins (like all other exchanges and custodial sites) has a wallet, but you do not. You have an account that keeps track of how many bitcoins they owe you. You do not get a private key because it is not your wallet.

When you trade on Localbitcoins using their escrow system (this is strongly recommended), the bitcoins are just moved between accounts. The seller's account is debited and the buyer's account is credited. They never leave Localbitcoins until you withdraw them to your own wallet.

The sender pays the mining fee, so when you send bitcoins to LocalBitcoins (or an address in any other wallet) you must pay a mining fee. It doesn't matter who owns the wallet or the addresses. Localbitcoins may also charge you a fee when you deposit bitcoins, but I don't know.

Quote from: camaxide on December 08, 2017, 04:31:03 PM

And last - I don't have an offline computer - but want to keep my wallet stored and backed up safely. As I dont intend to use it all the time but store it I'm thinking if I can simply compress the wallet.dat with a strong password, and store it at two locations thus meaning I both have a backup of it and also keep it away from access to others - will this work? There is no need for the wallet or so to be connected to the grid? If I later 5 years from now uncompress the wallet.dat onto a brand new computer I will still be able to access the bitcoins in the wallet provided I have both my private-key and my passphrase ? Do I need a second thype of key or address?

If you add a passphrase, then your wallet is encrypted and there is no danger of anyone stealing your bitcoins by obtaining the file. The only risk is someone obtaining your password, or somehow controlling or fooling the client when you use it.