What is self custody? When you buy your coins on an exchange, such as Binance. The coins are not really yours, under the crypto saying, 'not your keys, not your crypto'. They are in wallets controlled by Binance, and should they go bust, you are left with nothing - or a small % that may be recovered through legal proceedings. Self custody is where you transfer off exchanges into a wallet in which you have the keys and control, and therefore is much safer, assuming you are careful with your security. There are several types of self custody, this article will not be exhaustive but cover the most commonly used options - there are more available that you can find information on via Google.
Cold Storage
Cold storage is using an offline hardware wallet, you plug in (or connect via Bluetooth depending on the wallet) and can sign transactions with the device. The coins are not stored on the device itself, they are on the blockchain, however access to the coins is only via the device.
Ledger / Trezor are two of the main companies offering hardware wallets, it is recommended to only buy from the official site or an official reseller as if the device has been tampered with you can lose your coins when you send.
It is important when setting up the device you record your keys, (which is a series of words that you can regenerate your wallet from), somewhere safe - and not on any electronic device - not not take a photo, or put them in a word document or anything like that. Write them down off line and secure somewhere safe, for best security and fire prevention you can stamp them into a piece of metal.
Hot Wallet
There are plenty of hot wallets available, these are connected to the internet, and may be a phone app, computer program or browser extension. They are free, easy to access and generally secure - but only as secure as your computer and general security - so make sure your virus protection and malware scanners are up to date. Same applies as above for securing your passphrase/keys.
There are many available, some are multi-chain such as Trust wallet, some are specific to a particular coin or crypto. You can generally find information about which are best for a particular chain from the community, but beware of scammers. Never give seed phrases out to anyone, assume everyone messaging you is a scammer.
Scams
Most of the time when people lose crypto they haven't been hacked, they have been scammed. Basic measures you can take to avoid this are:
Never record your seed phrase on an online document/photo etc..
Never give your seed phrase to anyone
Check and double check addresses (the whole thing) before making a transfer and ideally do a test transfer first
Avoid giving any information to people who message you even if you think they are trying to help
Regularly scan your devices for malware and viruses
Emails from 'Trust Wallet' or any other wallet asking you to secure your wallet are fake, Trust wallet doesn't even have your email so how can they contact you?
Plus so many more.. be careful out there!