Central Udon Thani is small enough to walk, but it is hot enough that most people don't. Songthaews (shared pickup-trucks with benches in the back) run fixed loops through the city for 15–20 baht a ride. They are the quintessential Isan transport and the cheapest way to get between the lake, the markets, and the railway station.
Grab (the Southeast Asian ride-hailing app) works well in Udon Thani and is the easiest option for non-Thai speakers. Tuk-tuks and motorbike taxis are available but you will need to bargain. For the day trips out to Ban Chiang, Phu Phra Bat, or the Red Lotus Sea, hire a car with driver for the day — every hotel can arrange one for around 1,500–2,000 baht.