Bus - No. 3 Karori bus – from Lambton Quay, stops outside the Founders’ Entrance, Glenmore Street (just a short walk up to the Dell).

Car - There are a variety of entrances to the Botanic Gardens from around the surrounding roads. You can get a day parking coupon from a variety of places (service stations, dairies) before the day so you can make the most of your time at the Botanic Gardens. 
Access from various streets surrounding the Botanic Gardens include: 
Glenmore Street - various entry points: West Entrance, Pititea Entrance, Founders (main) Entrance
Mariri Road (Kelburn)
Glen Road (Kelburn)
Upland Road (Kelburn)
Salamanca Road (Kelburn) - East Entrance
Wesley Road (off Bolton Street)
Kinross Street (off Bolton Street)

Parking - The best chance at getting a car park would be outside the grounds on some of the streets listed above. Most of these streets have coupon parking areas.

Cable Car - A great way to enjoy Wellington city on the way to Kelburn. From here you can walk down to the Dell. Enter from Lambton Quay, Cable Car Lane, departs every 10 minutes. One way for an adult is $3.50, and $6.00 return. Under 5s are free.

Walking - The most popular route is from the Terrace through the historic and beautiful Bolton Street Memorial Park and takes about 15 minutes to the Begonia House.