Wellington Velodrome is a cycling park in Haywards, Lower Hutt, with a Google rating of 4.7.
A velodrome is a banked oval track built for fast, close riding, and most work with new riders through a short introduction or accreditation before letting them onto the boards. Track bikes have a single fixed gear and no brakes, so you slow down by easing your pedalling, which feels strange at first and is the main thing to learn. Regulars know that the banking looks steeper than it rides, that staying in your lane and holding your line matters more than speed, and that sessions are run in groups by ability. Wear a helmet, close-fitting clothing and shoes that suit your pedals, and ask about hire bikes if you do not own one. Spectators are usually welcome, and watching a race is a good way to learn how it works.
For riders in Haywards and around Lower Hutt, Wellington Velodrome is a very well-rated place to try track cycling.