Vijayawada is compact enough that you can cover the main sights — the temple, the barrage, Bapu Museum — in a single day on foot and by autorickshaw. Autos are cheap and ubiquitous; insist on the meter or fix a fare before getting in.
City buses run by APSRTC cover the wider city but are aimed at locals — confusing for short visits. Ride-hailing apps (Ola, Uber) work in the city and are the easiest way to get to the airport or out to Kondapalli.
For the Undavalli Caves and other sights across the river, an autorickshaw or a taxi for half a day is the simplest option.