This app was mentioned in 6 comments, with an average of 4.33 upvotes
Agreed all AT software is crud. Google Maps journey planner works well. And I use third-party apps for checking live boards, eg Auckland Transit
If you don't need to plan a trip, Auckland Transit works really well showing you times and routes. Even real-time GPS location of buses or train, although I don't really use this feature. I've used a few other alternatives but this one is the simplest and fastest to use.
It's not my "favourite" app, but I'll recommend some transport related apps:
Auckland Transit is awesome. Nice interface, clean, updated frequently. Some might say it's better than the official app
https://play.google.com/store/apps/details?id=com.dyancat.bus&hl=en
This one's great on android. https://play.google.com/store/apps/details?id=com.dyancat.bus&hl=en
Accommodation (and I'm going off what others have told me here): Central, that's backpackers budget I'm afraid. And private bathroom might be a stretch. Outer suburbs might net you a budget hotel. But honestly doubt 70 will get you much in Auckland. Best idea is to check booking.com or similar if you don't get good recommendations here.
Transport. Depends what you want to do and how far you want to go. And how long you're staying. Auckland Transport do day passes and month passes. Day passes let you use bus, train and most ferries as much as you like for those periods. Monthly passes are bus and train only. But you can buy separately month passes for ferries going different distances.
https://at.govt.nz/bus-train-ferry/fares-discounts/
Routes: use the planner tool. Route info for the buses is all over the place. Usually in pdf format.
https://at.govt.nz/bus-train-ferry/
Good free app to get is called Auckland Transit. It hooks into the AT database and gives you departure times. Has train and ferry tables too. Not sure if it's available for Apple. There's the official AT app called Track My Bus, but it only does buses. Bit pokey. But pretty accurate. And gives you a GPS location/ETA of most buses. There are other apps I haven't used in a while, but truth is AT is not great at building them.
https://play.google.com/store/apps/details?id=com.dyancat.bus
https://play.google.com/store/apps/details?id=com.ionicframework.athopapp712769
Car will be handy if you plan to visit places outside of Greater Auckland. We aren't like London where you can get trains to far out places. But there are regional bus services called Intercity and Mana.
Probably need more detail of your plans to help more.