Once your Bubble Dedicated Cluster is running. Your experience with Bubble will mostly be the same, except a few things:
You should still manage your account, create apps, update your credit cards, emails, etc. at bubble.io, as you would do before the upgrade. However, your apps are hosted on a subdomain (dXX.bubble.is). The first time you'll visit this URL, you'll be prompted to login with your Bubble credentials. When you'll edit your app (from bubble.io/home), you'll be take directly to the right subdomain. New apps will automatically be added to your Dedicated Cluster.
You can control the version of your installation in the Dedicated Panel visible at the top of the Editor (look for the fork icon). Clicking on it will show you if you're on the latest version or if you can upgrade. If you can upgrade, you'll be able to see what is new compared to your installation, and a button to upgrade. You can press this button and keep working (and refresh when the process is completed).