There are several methods of storing your coins these days. You can find people advocating for some and calling other methods bad. The various choices include leaving them on an exchange, putting them on a wallet on your computer, putting them on a hardware wallet or putting them on a paper wallet. If you want to leave it somewhere for a very long time, maybe a few years or maybe much longer (cold storage), where should you put them? Here, I am going to advocate for paper wallets.
First, let's put this in perspective. What you're wanting to do is generate and store two strings of characters, a wallet address and a private key to unlock that address. After you generate them and put your coins into the wallet you want to keep these two strings somewhere safe from destruction and you don't want others to get hold of the two strings. Which method of storing them would let you store them safely for many years? What method would you use if it were just two phone numbers you wanted to never lose?
The least secure method is, of course, leaving your coins on an exchange. Exchanges have been getting hacked, stolen from or closing their doors since the first exchange, The Bitcoin Market, opened its site for trading in March of 2010 and then 'scammed by Paypal' [1]. The list of closed exchanges since then is not a small one...MtGox, Vault of Satoshi, Crypto Rush, Mintpal, Gatecoin, Coincheck, Coinsecure, Cryptopia, FCoin, CryTrEx...just to name a few! Exchanges don't have the same guaranteed insurance that banks do, you are vulnerable to counterparty risk (some now have insurance, it's best to research if you want to use an exchange). If they go under without returning your coins there isn't much you can do as you never owned the private keys. These failed exchanges have lead to the famous saying "not your keys, not your coins'. I'm not saying don't use exchanges, just don't store more on one than you can afford to lose.
Next up is wallets you download to your online computer. These wallets will generate addresses for you and you get to keep the keys. With a wallet on your computer you can still use your coins, transfer them to other addresses, buy stuff with them...and you hold the keys. The danger here, of course, is that your computer could be hacked, someone could remotely watch your computer or could put a keylogger on your computer. There is also a danger of downloading a fake wallet with keys someone already knows or can access later; the general consensus seems to be that you should download wallets from the coin's GitHub page to avoid this. Reddit has a whole subject page on the matter at r/FakeCryptoWallets/ [2]. The subreddit is a whole other topic; if you're going to download a wallet it might behoove you to check recent posts, just to be safe.
The popular solution these days seems to be to get a hardware wallet. Hardware wallets will hold your private keys so you can store coins as well as sign transactions so you can use the coins. The two things you need are the hardware wallet itself and the PIN to use it. They range in size from the size of a USB drive to a slightly larger device that will just fit in the palm of your hand. It's definitely not as simple as that, as you also need it to communicate with a program as it will only sign transactions. I won't go into huge detail about them here, there are many people who will tell you about them [3,4,5] (99bitcoins.com/hardware-wallets/, en.bitcoin.it/wiki/Hardware_wallet, youtube.com/results?search_query=hardware+wallets, just for starters).
So why don't I like hardware wallets for storing coins for years or decades? The answer is two-fold. First, technology doesn't last. I have very few old electronic devices, particularly if they have a pc board in them. Most of my computers eventually went bad, so did my televisions, phones, answering machines, even simple stuff like toasters, refrigerators and other appliances. Electronics go bad, and you're paying money for that temporary electronic device. Will it be worth it? (You won't lose your coins if you keep the seed phrase for the device...now where are you going to put that seed phrase? On a piece of paper? In another electronic device that will eventually go bad?) If you store it on an electronic device, assume that electronic device has a shelf life.
The second reason I don't use hardware wallets for long-term storage is a bit further off...hardware wallets are just part of the technological arms race. Bank debit cards used to be considered very safe; now someone can wave a device over you wallet and capture the card's info and steal your money. Don't expect your hardware wallet to always be safe, it might not always stay that way.
Let me add, though, for shorter term coin storage, hardware wallets are awesome because you never have to expose your private keys to spend them. Hardware wallets definitely have their use.
The last wallet is paper wallets. The technology for writing doesn't go bad. For the price of a pen and piece of paper I have something that will last if properly stored. I have no electronic devices from when I was a kid, they have all been thrown out at this point. I have writings from my great-grandparents and it is still readable. Again, all I am trying to do is store two character strings for a very long period of time. If I want added security I can write the two character strings down twice and store them in separate places.
The biggest issue I have with paper wallets is making sure I generate the keys safely. All my paper wallet keys are generated on a computer that has no internet. Even if someone got malware or keyloggers onto my computer, it would never go anywhere. Two methods I will recommend are either a wallet with BitKey, such as the ELECTRUM wallet, or using the browser on your offline wallet to go to bitaddress.org and letting the java code generate your wallet. [Note: once youre coins are on a paper wallet you will want to sweep the entire amount onto a wallet to use it.]
For a more thorough read on generating safe keys I recommend starting with the wikipedia page en.bitcoin.it/wiki/How_to_set_up_a_secure_offline_savings_wallet, as well as following that pages links and references[6].
1. Bitcoin forum post, "Recent Events At Bitcoin Market", bitcointalk.org/index.php?topic=12678
2. Subreddit on fake cryptocurrency wallets, r/FakeCryptoWallets, https://www.reddit.com/r/FakeCryptoWallets/
3. "Hardware Wallets Explained, Reviewed and Compared", 99bitcoins, https://99bitcoins.com/hardware-wallets/
4. 'Hardware Wallet' wikipedia page, https://en.bitcoin.it/wiki/Hardware_wallet
5. Sample youtube query for 'hardware wallet', Youtube, https://www.youtube.com/results?search_query=hardware+wallets
6. "How to set up a secure offline savings wallet", wikipedia page, https://en.bitcoin.it/wiki/How_to_set_up_a_secure_offline_savings_wallet
