There's a handy video you can watch on Getting Started in VS Code with Salesforce DX.
First things first: it's important to note that VS Code places extension prompts at the very top of the screen, instead of the more usual center of the screen. You'll need to pay attention to these dialogs, which are not immediately obvious. Once you get used to this, you'll find that Code is not that hard to use, it's just different than most other IDEs.
To make a new DX project, choose the Create Project from Manifest command, and you'll see a small text area at the top that says "Enter project name (Press 'Enter' to confirm or 'Esc' to cancel)." Type in the name you'd like to use for your project (it can be any reasonable text).
Next, you'll get a Folder Selection dialog. Choose a place you'd like to save your project. It's helpful if you've already created a workspace folder to store all your projects in, such as
c:\users\your.name\workspace\. Once you select the folder, you'll see a brief progress dialog, after which your project will be ready.
Once you do this, you'll probably want to Authorize a Dev Hub org. There are steps you must follow to enable Dev Hub functionality before you can use this command. The Dev Hub can be a Developer Org or your Production Org. This is optional, but if you do not do this, you won't be able to create scratch orgs. If you're just using a sandbox or developer edition, you can choose Authorize an Org instead.
Again, pay attention to the dialog at the top of the screen. You'll want to enter an alias (strongly recommended), which is simply a nickname for the org you're authorizing. After that, you'll be prompted to log in through your default browser. Once logged in, you can go back to Code.
If you're using Scratch Orgs, you can create a new Scratch Org, which will take a preferences file (one was created in your project automatically) and an alias. After about a minute, you'll have a new, empty org. Everything else from there is just development; you can use the DX commands to create new classes, pages, components, etc, or you can manually create the source files if you prefer.
Or, you can log in to the scratch org, make changes there, and use SFDX to pull changes down to your computer. Choose whichever works best for you. You'll notice when you create a scratch org that you have no username or password; this is fine by default, as you can just choose "Open Default Org" to go to the scratch org you created. Note that not all DX commands and options are exposed, so you might need to use the Terminal from time to time.
To do this, just click on the Terminal tab, and you can use the sfdx command with all its options. You can check the Salesforce CLI Command Reference for a full list of commands and options available. You'll also want to read more about the configuration options for your scratch orgs and other settings, but this should at least help you get started.
Also, feel free to Get Started with Salesforce DX over on Salesforce Trailhead for more DX basics. These are not VS Code specific, but will help you understand the intended development cycle.