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.
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.
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?
Hope those are questions easily understood - please reply if you know the answer, or ask away if there are some of my questions that are hard to understand

Not really, just make sure the environment of bitcoin core is clean and secured against any viruses etcetera. ( although this still shouldn't matter if you encrypted it )
You should indeed be fine, although i wouldn't really recommend compressing it, because there's really not that much space to gain --, and it might become a bit of a hassle.
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.
If you buy them from say, Coinbase, you will need to pay a network fee above the fees you already paid for your purchase, if you want to transfer them to a wallet outside their ""Network""