Your best bet is to take a simple city tour in BKK for 2 days and reserve some days for shopping in BKK before leaving.

You may want to check their train service in Thailand from BKK to Chiangmai and stop at whichever town/city which you are interested in before reaching Chiangmai. The old cities of interst would be Ayutthaya and Sukhothai i believe.

You may check the budget airlines too. They prominent ones are AirAsia and Nok Air