Hello @Eric Wivczar , Welcome to the Microsoft Q&A forum.
Based on your questions above.
From my understanding I can use any private ASN I choose.
You can use both 2-byte and 4-byte AS numbers. You can use a private AS number for this peering except for the number from 65515 to 65520, inclusively.
And any VLAN ID I choose as long as that ID isn't associated to another VLAN on my network.
Yes, a valid VLAN ID is required. Ensure that no other peering in the circuit uses the same VLAN ID. For both Primary and Secondary links you must use the same VLAN ID.
Once the circuit is established I can then use that info with the proper IPV4 addresses to configure my local Router.
I might not have understood this part of the question correctly. To configure private peering, the ExpressRoute circuit must be in a provisioned and enabled state. A pair of subnets are also required that aren't part of any address space reserved for virtual networks. One subnet will be used for the primary link, while the other will be used for the secondary link. From each of these subnets, you'll assign the first usable IP address to your router as Microsoft uses the second usable IP for its router.
These details are currently documented here, going through the pre-requisite section can also help here.
You can go through this documentation to verify your private peering connectivity.
Hope this helps. Please let me know if you have any additional questions. Thank you!