Choosing your bitcoin wallet is in the same value of owing them, it is a matter of privacy, freedom and security.
[1] Raw Wallet: storing your bitcoin offline on a paper containing all the information to generate your bitcoin private key, e.g. barcode than can be scanned with any software wallet you desire.
It is your strongest cold storage / physical backup and trusting yourself by yourself!
[2] Hard Wallets: similar to a raw wallet, but more as a hardware solution oriented.
They are safe and user portable as well.
You can find many trusted hardware wallets usb-dongles such as Trezor, Ledger
[3] Soft Wallets: instead of using a hardware wallet, it is here using a software wallet to manage you bitcoins, you have to make sure about the trust and reputation of the wallet. e.g. Electrum wallet does not own or save your private key it uses your pass phrase/password along side to generate it when you need it and never send it on the internet, operations are done on your pc/mobile client side https://en.bitcoin.it/wiki/Electrum
[4] Online Wallet: it is a web wallet, that own your private keys and manage your account. you have to take care about the reputation and safety of wallet. one of the best online wallets https://blockchain.info
[5] Online Exchanges: more like a web wallet, but less secure. they may offer you a bigger range of wallets and currencies wallets support to trade, exchange, deposit and withdrawal. more preferred to be used for trading and exchange than a personal wallets.
Biggest and most secure Exchange-Online Wallet is https://coinbase.com
