Currently, the easiest way to access Handshake top-level domains is by using Handshake-compatible DNS resolvers like nextdns.io. It was designed to be a quick solution to let people experience Handshake without having to require any of the heavy lifting. You can read more about how to access Handshake domains in this tutorial.
The ideal way to access Handshake top-level domains is to change your OS settings to point to the Handshake light client resolver.
The long term goal for Handshake is to get browser buy-in so Handshake names will work out of the box without any additional extensions or steps required. Brave has already expressed interest in Handshake and will likely be the first browser to adopt it. Additionally, popular web3 browsers have expressed interest in supporting Handshake too.