To buy Litecoin directly, it depends where you live. If in the US, you can buy with Coinbase directly from your bank account to Litecoin. If elsewhere in the world, you may have to buy Bitcoins first, then use Poloniex, or Bitfinex to convert BTC to LTC.
To store Litecoin, if you plan to invest more than the equivalent of $1000, I would strongly recommend any of the hardware wallets on the market (Ledger, Trezor, KeepKey). All three work with Electrum-LTC (electrum-ltc.org) - the private keys remains in your hardware wallet, and the Electrum SPV network does the checking and broadcasting for you.
Remember, the Litecoin resides on the Litecoin blockchain (in the cloud), your "wallet" only holds private keys that allows you to transfers LTC from your adresses to someone else.