I would avoid downloading wallets or creating backups on computers you don't fully trust. It's a bit like logging into your bank account online. Would you trust a public computer not to have a keylogger?
Generally you should download the wallet on your home computer, password encrypt it, and copy backups to flash drives and external harddrives. By far the greatest risk to your bitcoin is not hackers but actually losing the files by accident. So more backups the better.
I'd link you to a 100% secure guide to creating and backing up a bitcoin wallet but it's a bit technical so for now I'll hold off.
You can also create wallets through the major websites and they'll back it up for you. Now if you want to be in control of your own safety you would do the backups yourself.
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