In the Exchanges, you have to do research on this, all the exchanges are not safe, you have to look at the most recognized and be careful.
When you have Bitcoin on an exchange, you don't have your private keys.
Not Your Keys, Not Your Bitcoin
Once you have bought your bitcoin or part of your Bitcoin you have to send it to your wallet, you are your own bank and it is your responsibility to protect them.
Wallets:
- Desktop:
Bitcoin Core
Electrum
Exodus
- Hardware:
Trezor
Ledger
KeepKey
- SmartPhone:
Bitcoin Wallet
Mycelium
- Paper Wallet:
BitAddress
