There are many forms of transportation that can get you to the places you want to visit. Here are your options especially when you are a budget traveler.
Ride-sharing services: Uber vs. Grab
Coming from the US, I already had Uber installed on my phone. So it was handy when we didn't like a price quoted by a tuk tuk driver. Lyft is basically non-existent in Southeast Asia so I suggest removing that useless space-taker and install Grab instead, Southeast Asia's Uber. In fact, Grab was cheaper than Uber most of the time especially in Thailand. For example, it costs 150 baht on Uber while it costs 75 baht on Grab going to the same destination.
On Grab, you also have the option of paying in cash if you do not want to pay by card. Additionally, by using Grab, you earn points and with the points you can redeem discounts like 10,000 dong off a ride in Vietnam.
Currently, UberMoto and GrabBike are banned in Thailand due to legal issues. But once the ban is lifted, I suggest trying it out at least once. If not, remember to try it out in Vietnam!
Local buses & subway train in Bangkok
Since my phone carrier is T-Mobile, I had the luxury of enjoying free 2G data while I was in Thailand. So finding a bus stop using Google Maps was not a difficult task. If you like many other travelers do not have data, I suggest downloading MAPS.ME-> life-saver.
Local Buses
Anyways, local buses are the cheapest option to get around Bangkok. It ranges from being free (we luckily got on one of these) to costing only 7 baht (~$0.20). There is no AC on the bus but there will always be a breeze whenever the bus is in motion. There are no APPs or stop announcements that will tell you when to get off so you would have to track it on your phone. In fact, the bus route is hand-drawn out which shows how antiquated the bus system is compared to the US and China.
To get off, there are signal strips close to the ceiling. Press that to tell the driver you are getting off. If you are going to a popular spot, many people will be getting off to so you don't have to do much but walk towards the door.
Subway
The subway system is more modernized with a readable map of the route that announces the name of the station every time it makes a stop. It costs around 20 to 40 baht depending on which stop you want to get off at. I suggest living at an accommodation close to a subway stop because you have more bargaining power with a tuk tuk driver.
You will need coins to purchase a ticket - walk up to a customer service rep and tell them where you are heading. They will provide you with enough change to buy the ticket at a machine.
Tuk tuks
Tuk tuks are everywhere in Thailand. It is one of the most convenient and fastest way to get around. The catch is there are no fixed prices or a metered price so you would have to do some bargaining and I suggest doing some research and asking around to figure the correct range. Depending on their mood, the drivers tend to inflate the price for foreigners so definitely bargain. Bargaining here is not as aggressive as in China but you should be able to bargain it down to close to the usual price.
Even though many Thai drivers know English, it would be easier if you show them a place on the map in Thai by zooming in. Underneath the English, a set of Thai characters appear - it will make your life so much easier!
Songthaews
These are red pickup trucks mainly in Chiangmai. They are actually cheaper than tuk tuk since they can take up to 10-12 people and even more by having people standing and holding the rails in the back. It should costs around 15 to 20 baht per person at a location 10-15 minutes away from the city center where the night market is (~30 minutes of walking).
Walking
I leave the best for last. The old way of getting around is walking. By walking, you can easily experience more and discover more. We took a local bus to Chinatown in Bangkok but we walked back from Chinatown to Khaosan Road. As we walked back, we passed by many food stalls, a street filled with automobile repair services, a lively park, a temple, and the Grand Palace. We encounter some very friendly guards at the Grand Palace entrance!
On our last day in Bangkok, we walked again and discovered a very modern area filled with nice cafes and restaurants. What I want to say is if you can walk it, then do it especially when you still have the ability to do so!