How to send/receive Bitcoins from cold wallet?
"Is there a way to do this (send / receive) without needing to use my private key on a computer connected to the internet?"  


Yes, search for 'offline wallet'. The basic idea is the following:


download the 'offline wallet' webpage you liked
you must be online to fill blanks like addresses, amount and fee
sign the transaction while offline and copy it
go to a website that broadcast transactions
paste your raw transaction already signed and push it


In my opinion, this is the safest way for non-developers to do bitcoin transactions.