All the suggestions above are perfectly good. However, when you have a house that is entirely painted, the options are so endless that it's easy to get stuck. I'd advise either finding photos of comparable homes in colors that you really love, and reproducing a similar palette on your home… or upload a picture of your house to a colorizing site, and play around with various combos. Sherwin Williams has a Color Visualizer that is decent. I don't mind a white house, but feel that the black trim is a bit harsh. On the railing especially- it feels like it is outlining itself too much. Reduce the contrast between the body and the rail and it will enhance the house rather than overwhelm it. You have a lot of choices, so just experiment and go with what resonates with you.

An outdoor room is like any other room. First decide what you want to use it for. For example, my back yard has a couple of rooms. The "dining room" is a table and 4 chairs under an inexpensive ($150) 12 ' X 12' Home Depot gazebo . The gazebo gives a pleasant feeling of enclosure and privacy when we sit at the table, and it also keeps the table clean and dry. We have a big hydrangea shrub right at the edge of this area to mark its edge. It's pleasant to sit so close to something blooming. The "living room" is 2 comfortable chairs and a love seat around a coffee table. We entertain there and pull the dining room chairs over if there are more than 4 of us. We also sit there to read in the evening. There's a smallish tree right beside it so it's pleasantly shady. The "kitchen" is the grill and a small table to set things on, and is closest to the kitchen door. Once you have decided how you want to use the space, you'll need to decide whether you want a new "floor". We have a low L-shaped deck that has the three rooms. Or you may decide on a concrete or slab patio for all or part of it. Farther from the house is some grass, and then more big shrubs (evergreen and blooming - little maintenance). And a little raspberry patch because it's fun to eat them right from the bush. The landscaping really comes last, once you have decided on the functions of the area. Don't start planting without a room plan. You may want, for example, to put one of your rooms right against the fence or wall, and if you've already planted shrubs or made beds then they will be in the way. You want to plant trees and larger shrubs so they are useful as the walls and roof of your rooms. Be sure to check the mature size of your plantings (it's on the tags) so you don't end up with branches hanging over your roof or over your neighbor's property. Don't make the mistake of putting in so many beds or plantings that you don't have time to sit and enjoy. A few low-maintenance shrubs that are planted close to where you sit are just as enjoyable as big swaths of flower beds that need weeding, annual bedding plants and maintenance (unless gardening is a hobby that you enjoy spending time at.) Enjoy your new home!

I don't know how hot it gets in Auckland in summer (I've only ever been to NZ in winter!) but if it gets hot, then go for a pale colour. Dark roofs should be kept for cold places!

I agree with havingfun, thicken the supports and have something growing trailing up them. I didn't even notice there was a pergola until the close up. Right now it blends in too well with the house color, consider painting it a different color. I don't know your usda zone but classic climbers for a pergola include clematis, wisteria, grape https://www.houzz.com/photos/santa-ynez-valley-mediterranean-porch-santa-barbara-phvw-vp~2485412 or go for a modern look You can also hang string lights, planters or something from the beams. Also agree lighten the front door.