The desktop (/local) wallets download the whole transaction history (yes, of every single Bitcoin in the world), so it will take a while for the first sync ^^
I think the one for smartphone has other ways to access the transaction history.
The safest place to store Bitcoins for a long time are offline wallets or paper wallets.
The easiest way is imo creating a paper wallet with this website: https://www.bitaddress.org/
There are some things you should know:
Keep the private key safe! It's like the key to your safe!
The Bitcoin adress is the adress you can use to put any amount of Bitcoins into this safe.
Important: If you want to take any amount of money out of the safe, take it all and don't use this paper wallet again! Create a new one!
i remember when i was playing around with my android wallet just to see how it works i exported the keys or something like that and saved the file to dropbox then i formated my smartphone , downloaded the android wallet app and just imported the keys
is this similar what you;re talking about with the paper wallet?