Downloading a wallet means, downloading the whole blockchain with it, if you will be using a wallet such as bitcoin core. You can download a wallet from the Google play store and make a backup of your private key which you could store in an external hard drive.
    Edited on August 30, 2017
    basically the wallet is a piece software that can be downloaded somewhere and run from the location. So, supposing a laptop you can download to an external drive or USB stick and run from the drive only of it is connected (so both the software and the keys are on the external disk).

    a similar but a little bit more secure version is the so called hardware wallet

    All wallets for blockchains will be free. It sounds like you’d benefit from these guides aimed specifically at total beginners in crypto. They’ll take you step by step through buying and storing a whole range of coins: