I have started off by downloading a bitcoin wallet - "bitcoin Core" and it's in the process of synchronizing . Is this a good wallet? Is it safe from hackers, etc? I also want to be able to store my bitcoins offline on a harddrive - I think it is called 'cold-storage' - This stops malware/viruses taking your bitcoins. How do I do this? Do I need a separate wallet? Can I store my bitcoins both on my laptop and offline hardrive or can I only store it on one?
Bitcoin Core is a great wallet.
Your bitcoin is safe as long as you are managing it right (eg. don't download suspicious files, encrypt the wallet file and make a few backups, etc.)
The easiest way IMO is to create a paper wallet. Just go to bitaddress and save the page as html. Disconnect your computer, create the wallet, print it out, laminate the paper and store it in a safe place.
It is okay for you put the same wallet on your (online) laptop and your offline hard drive, but then it will defeat the point of making a cold storage.