You definitely want your own wallet address, rather than using an exchange.

You can set up your own node, it's not too difficult but does require a bit of RAM or at least swap memory. And some time to fully synchronize the blockchain.

Alternately, you can run simplewallet without your own node by syncing it using a public node, such as the one I maintain.
Instructions here: https://www.hiddenjadestone.com/2017/06/24/aeon/setup-aeon-wallet-using-windows-light/

You can even just run simplewallet without any node connection, and have it generate a wallet address. Then you mine to that address. In order to verify that you received your coins, or to transfer them, you will at some point have to sync your wallet with a node.