How do I update Cmder to use PowerShell Core?

Use

*pwsh -ExecutionPolicy Bypass -NoLogo -NoProfile -NoExit -Command "Invoke-Expression '. ''%ConEmuDir%\..\profile.ps1'''"

I currently use Cmder for this, which is based upon ConEmu. There are already two PowerShell tasks and I assume that I can update them to point to my local install of PowerShell Core (in C:\Program Files\PowerShell\6.0.0).

You are using the wrong executable.

The binary name for PowerShell Core has been changed from powershell(.exe) to pwsh(.exe). This change provides a deterministic way for users to run PowerShell Core on machines to support side-by-side Windows PowerShell and PowerShell Core installations. pwsh is also much shorter and easier to type.

What's New in PowerShell Core 6.0

Additionally

You can launch PowerShell via the Start Menu or $env:ProgramFiles\PowerShell\pwsh.exe

Installing PowerShell Core on Windows