Or would I need a SmartCard "writer"
Most (all?) smart card readers can write values to the card. All you should need is the smart card. It's worth noting that you probably want a cryptographic smart card. They tend to cost more than $10. There are other smart cards that cost less but don't do the encryption on the card (which makes they much less secure). You will also want a card / drivers that will work with your operating system. Setting up the drivers can be a bit of a pain. There are also different kinds of certificates. For example, there are PGP keys and there are x509 certs for PKI. You will want a card / reader that will work for what you are doing. Do some googling and see what cards other people have used on your OS.
Would it be wise/possible/practical to buy a single SmartCard,
transfer my private key on that (somehow), and use it as my primary
private key storage?
If you are concerned about security, it is a very good idea. Computers have many avenues of attack while a smart card has far fewer. Using a smart card can keep your private keys safe even if your computer is compromised. Just make sure you do some research on your reader and the card you are looking to buy to make sure it will work with your OS / application.
Another note: although you can transfer an existing key to a card, it is better to generate a new one in a safe environment. You don't know if your existing one has been compromised before you put it onto the card.