Yes, most of the bitcoin wallets are free, unless it is hardware wallet that you are talking about. There are a few types of bitcoin wallets, namely web wallet, mobile wallet, desktop wallet, paper wallet and hardware wallet. We have written a beginner guide to bitcoin wallet, if you would like to learn more, you can take a look at the following article:
Bitcoin Wallet for Beginners: Everything You Need to Know

Fees might be involved if you are sending bitcoins to another address but those fees are not imposed by the wallets but are meant for the miners to process your transactions. You are always advised to store your bitcoin in a cold wallet ( offline wallet ) instead of on exchanges ( hot wallet ) as there is always a risk of hacking for hot wallet. But you should also beware not to lose or misplace your cold wallet as that would mean that your fund is lost forever. Storing your fund in a hardware wallet like Ledger Nano S or Trezor Wallet is highly recommended if you can afford it of course. If you are new to cryptocurrency, we would suggest you to take a look at the articles in our beginner’s guide section. ( http://cryptoverze.com/category/beginners-guide/ )

Should you have any other questions, feel free to ask away, we are always here to help. 🙂