You don't need to transfer between forks or anything of the like, the only difference for you is that you now have two types of coins. Eg if you had 21 ETH before the fork you now have 21 ETH on the hardfork chain and 21 ETC (ethereum classic is what they call it) on the hardfork opposing chain.

Besides there are light clients that don't require the entire chain to be stored on disk, there are even some that run entirely in your browser like https://www.myetherwallet.com

Now if you want to interact with the hardfork opposing chain things gets a bit more difficult because it doesn't have much dev support and as far as I know there are no light clients supporting it.