If you’re going to Kagoshima and just intend on staying in Kagoshima-shi, I do not recommend renting a car or catching a taxi but instead, I recommend using the public transportations (bus, trains and trams). There are day passes you can get on trams and at the Kagoshima Chuo Station which allow you to catch certain buses and the city trams for free. I recommend those passes. You just show them to the driver and you’re good to go. Initially you have to scratch the date on which you would like to use the pass. And it’s that part of the pass that you would present to the driver.
Also, if you’re unsure about the bus’ destination, you can hurriedly ask the driver if he’s going to wherever you’re going. Just be mindful though that during rush hours, early mornings (when persons are going to work), lunchtime, and afternoons (when people are getting off from work), as well as rainy days, festival days and holidays (including weekends), the trams and buses can be really crowded.
Be respectful of the priority seating (for pregnant, handicapped and senior persons). Also be aware of your big bag pack, my rule of thumb is to turn my bag pack and have it on my chest so that I’m more conscious of it and not allow it to hit anyone, especially when standing in the trams.
If you have a stroller or a wheelchair, every other 2-3 trams, you would see the more spacious and handicap friendly trams which are meant to accommodate strollers and wheelchairs. There is one set fee for adult no matter where you got on, it is currently ¥170, and for children 6 and over (I think) it is ¥80. And should you wish to get off, just press the button when you hear the message telling you what the next stop is, if that’s your stop.