Ethereum wallet - is, first of all, two things:
1) Your public key - it is exactly Your Ethereum wallet number. No need to hide it.
2) Your private key - an analog of the usual password. It is your personal. The loss of a private key means the loss of the Wallet's control. If someone seized your private key, he seized your money, and you will not be able to change it for another private key.
We never ask you private keys
There are several Ethereum wallets. The most popular and simimple in use - MetaMask and MyEtherWallet.
Features of each of them are written below.
2) Trust Wallet
It is the web version of the Ethereum wallet, therefore it is very easy to use and suitable for smartphones.
On use of MyEtherWallet, you do not need to install plugins and applications. All transactions are made directly from the site MyEtherWallet.
Detailed at MyEtherWallet FAQ
Web version: https://myetherwallet.com/
Impornatnt! If you have registered Ethereum wallet on any service, You can always continue using with another one.
For example, if you registered Ethereum through MetaMask, you can continue to use the SAME wallet through MyEtherWallet and contrariwise.
All that You need - Your private key.