Amex used to have a progress meter (like other banks) in the past for selected customers, but seems they are gone now. So now the easiest way is to call or message amex support and ask them.
Any way to tell how far along I am on progress to the 6k spending to get the sign up bonus?
Add your past payments to your current account balance. Subtract the annual fee paid (if any), as well as anything that counts as a cash advance, rather than a purchase.
All of this information should be viewable after logging into your account.