Most guidebooks or blog posts about Kenting National Park will tell you to rent a scooter, but you can actually do without. That leaves you with two main options: walking and taking the bus. All this is assuming that you are based on Kenting Street, which is a pretty good idea since it is a very central location.
If you are not afraid of it, a few places are in walking distance of Kenting Street. Usually, there is enough space for pedestrians to walk on the side of the roads. It can get a little narrow sometimes, so be vigilant, but overall it's very doable.
There are a few buses in Kenting. The bus 9188 goes from Kaosiung to Eluanbi (stoping at Hengchun, Kenting Street and Sail Rock, among others). Buy your ticket directly from the driver, tell him your destination. Most drivers do not speak English but will understand the name of your destination. Have some change, because you need the exact amount, drivers don't give change back. For example, Kenting Street to Eluanbi is 25 NTD (around 15 minutes). If the driver gives you a ticket, you'll have to give it back to him before leaving the bus. It's best to follow the journey on your phone's GPS and press the stop button when close to your destination.
Note that the 台88 bus is a different bus than the 9188, many people get confused.
There is an app called "Bus+" available both in Chinese and English, giving you in real time the waiting time and the buses position. It's pretty straightforward to use once you know the bus number. Also useful to get the bus stop names: Android link, iOS link.
Sometimes a bus seems to disappear, give it a few minutes, it often happens when the bus leaves its starting point and has not yet reached the first stop. But if the bus does not re-appear after a while, it might have been cancelled.
Be in advance if you want to take a bus, it's not rare for a bus to pass before its scheduled time.