Transiting through Beijing and Shanghai

If I understand correctly, your route should be Hawaii - BJ - Shanghai - Taiwan, right ? Well then, to transfer at BJ, here is the procedure:
1: Inspection & Quarantine, Exit and Entry Frontier Inspection
2: Baggage Claim
3: Customs Inspection
4: Going through Transfer Formalities
5: Security check
6: Waiting & Boarding

At Shanghai, here is the procedure:
arrival - domestic arrival access - luggage claim - confirm boarding pass in transfer counter - Inspection & Quarantine, going through immigration and security check - international flight waiting

This is just the regular procedure. As you will use air china for the whole trip, things would be much easier to handle. I believe the staff will arrange your transfer well.