In 2014 I was flying from Don Mueang Airport and it always brings back good old memories of the old days when traveling to Thailand. It was in 1985 that I landed for the first time on Don Mueang Airport and walked over the tarmac. No gates, you just walked to a main entrance door. Total chaos is what I remember. But that was 1985! A lot things changed since 1985, even the airport code for Don Muang! Until the opening of Suvarnabhumi Airport the airport code was BKK and now it is DMK.
Don Mueang airport is actually a militarily airport (as many all airports in Thailand). When in 2006 the new airport Suvarnabhumi was opened "all" commercial flight were moved from Don Mueang to the new airport Suvarnabhumi. But due to the not so clear design / building of the new airport it came very quickly known that the new airport had a capacity problem. So it took them 2 years to move all flights to the new airport (2008) at what time it did seem that Don Mueang would become a abandoned airport. But TIT .... the capacity problems of the new airport were not solved and domestic flights were moved back to Don Mueang. So now only Bangkok Airways and Thai Airways are flying domestic flights on Suvarnabhumi but all other domestic flights are on Don Mueang. And now even also international flights from budget airlines to Malaysia, Singapore, Cambodia, Myanmar, Indonesia, Hong Kong and Macau are using Don Mueang airport.
The Domestic part of the airport is still old style and in desperate need of renovation. After check-in you proceed to the gates area where are fast-food restaurants and some shops. When arriving you can take a taxi at the taxi cues for what you pay 50 Thai baht extra (what of course is just legalized taxi mafia money). The International part of the airport seems to be renovated and there are plans for renovating the Domestic part also.
There are shuttle buses going to the city and to Suvarnabhumi Airport. But be aware that transfer to Suvarnabhumi Airport will take about 1 hour. That is not included exiting the airplane, picking up your luggage and waiting for the shuttle bus to leave. So all together you should have at least 2 hours extra for transfer from Don Mueang Airport to Suvarbabhumi Airport. That means you should have 3 to 4 hours between 2 flights at different airports.