you would download litecoin core onto your computer and use a wallet such as mycelium or jaxx or Electum for your cell phone...I recommend only using cell phone wallets for small or temporary storage...you need around 25 GB of Drive space to store the blockchain and it take about 2 days at most to download on a cable connection...longer on slow wifi
you cant use a core wallet right away because it has to sync with the rest of the network first...where as wallets like the ones you can use on your phone use third party servers who are using core wallets and issuing address' using 12 word seeds...which are NOT private keys...they are backup phrases
a private key is the only way to truly store a Bitcoin or Litecoin...because if a seeded wallet server goes down..all the coins that are relayed to the address' it generated go with it...
with core wallet you can save a backup file on a usb flash drive which will recover your wallet if your computer ever crashes...make sure to always use a spending passphrase too /aka wallet encryption
go to litecoin.org and you will see all the wallets available for download for your devices