Is it better to buy my airlines tickets (Chiang Mai-Yangon & Bangkok-Siem Reap) before leaving or when I get to Bangkok? And if so, does anyone have a good tip for a travel agent or airline office I could go to?

As most airline tickets in SEA are now priced in US dollars (converted back to local currency if you buy there), you don't have much to gain either way. If you buy in Bangkok, don't hand over cash until you see them printing tickets for you - some v. dodgy travel agents there.