Technically, you are not saving your bitcoins in a wallet. They are stored in nodes which contain information about every single Bitcoin transaction. To send your bitcoins to someone you have to sign the transaction with a private key. Every address has its own, unique private keys which won't work for any other address. There is no risk in storing Bitcoin as long as you are the only person who knows the private key. I would recommend you Electrum and Bitcoin core if you don't want to spend any money.