If your solution has a gobal.json file, you can either delete it or update the sdk version over there to the latest. At the point of this answer the latest .NET version is 6.0.0 with the sdk version as 6.0.100 (Latest sdk version could be found at the dotnet core site where you downloaded your sdk)

{
    "sdk": {
        "version": "6.0.100"
    }
}

Where to confirm your sdk version from your download

After you have done that, 'Unload' and 'Reload' your project and the latest .Net Core target should show up in the dropdown.

During Visual Studio setup, you need to select the ".NET 6.0 Runtime". As can be seen in the screenshot, this option not only includes the runtime itself but also "templates for developing [...] .NET 6.0 applications".

You can modify your installation by starting "Apps & Features" from the Windows start menu or by selecting Tools/"Get Tools and Features..." from the Visual Studio menu bar.

I wonder if this is the same problem I've been struggling with - you can't change an existing project that is not in the 'sdk' format to net 6. But there is an upgrade assistant that will convert it for you.
See https://docs.microsoft.com/en-us/dotnet/core/porting/upgrade-assistant-winforms-framework

You have to install the upgrade assistant with

dotnet tool install -g upgrade-assistant

After installation, you can either analyze your project

cd "C:\your solution path"
upgrade-assistant analyze yourProject.csproj

or run the upgrade by

cd "C:\your solution path"
upgrade-assistant upgrade yourProject.csproj

The assistant will guide you through the steps.