Short answer
Test complete business processes end to end with your own converted data, run payroll in parallel until results match, and have the people who will use the system sign off on each scenario. You are ready when your staff can perform their real work in the new system, not when the vendor says configuration is done.
Unit testing tells you a screen works, but what you need to know before go-live is whether a requisition can turn into a purchase order, a receipt, an invoice and a payment, and whether that all lands correctly in the general ledger and the budget. I would build end-to-end test scripts from the same scenarios you used in your demos, run them on converted data rather than clean sample data, and include the integrations with your banks, timekeeping, permitting and utility billing systems.
Payroll needs its own discipline, so plan for parallel runs where you pay employees from the old system and calculate the same payroll in the new one, then reconcile every difference down to the cent until two consecutive runs match. I also like to include reporting in testing, since council and auditors will judge the new system by its reports, and a missing year-end report is a bad surprise in the first month.
Most importantly, make sure the testers are your own staff rather than the implementation partner's consultants, because user acceptance testing is also the best training you will ever get. The Start at Zero Navigator includes a go-live readiness checklist, and it is a free download.