Australia and New Zealand

Sounds like an wonderful trip!

I've been to Sydney twice, both for a week-long stay. I don't rent cars when I am on an international trip but rely instead on public transportation, Uber/taxi, or a day tour with a touring company. Public transportation is quite good in Sydney and includes public ferries that can also take you to spots outside of Sydney proper, such as to Manly Beach.

I took the train (about two hours from the main Sydney station) to Katoomba, home of the beautiful Blue Mountains, both times that I was there. You can do some light hiking and afterwards wander around the cute little towns of Katoomba or Leura.

On one trip I took an all day wine-tasting tour to Hunter Valley, which was a lot of fun and a nice way to see some other areas outside of Sydney.

For help with routing back home, we would need to know where home is. As for my two trips, I flew from Tucson, AZ to LAX and then took an American Airlines flight nonstop to Sydney. My daughter met me in Sydney on one of the trips and she flew nonstop from Dallas/Ft. Worth to Sydney on Qantas.

Greetings Anne,
What an exciting trip you have ahead! We love Sydney and have been there before. I would suggest you definitely get passes for both the Red Line and Blue Line tourist busses. They are great way to see different neighborhoods. We did that and loved it.
You may also want to consider taking a tour of the iconic Opera House - you learn so much about it and it's fun being in the Circular Quay (pronounce "key") port area. We also enjoyed a walking tour of The Rocks adjacent to Circular Quay. You find those online. They are both informative and fun.

You mentioned you are both walkers - so are we! I highly recommend walking across the Sydney harbor bridge-- such a great experience! On the other side you could make your way to Milson's Point - a park with great views back across the water of the downtown area. Then, while you are on the other side of the harbor, take in the Taronga Zoo. It's a delightful experience. You could get public transportation back to your hotel.
Another fun excursion is to take the ferry to Manly Beach. There's a ferry that leaves directly from Circular Quay --
Wharf 2, Side A and arriving at Manly, Wharf 2. Services depart every 10 minutes, and operate every day. The journey takes approximately 20 min.

For a day trip, consider a visit to the Blue mountains. There are a variety of companies that offer the day trips and will pick you at your hotel. I see another traveler recommended taking a train to the area - even better!
We did not rent a car. We just did not feel comfortable trying to get accustomed to driving from the right hand side.

For a budget accommodation consider Mariners Court hotel. It's a bit of a walk to the CBD but a great deal - just $120 a night and includes breakfast - it's near the Woolloomooloo Wharf complex. A friend stayed there and said it was fine. If you wanted to be closer in to the CBD you could consider Rendezvous Hotel / The Rocks -- that is in the Rocks area, adjacent to Circular Quay. It runs around $270 (USD) a night but the rooms include a kitchenette and a balcony.

Below is info on the routes of the Blue and Red line tourist buses.
Hope this is helpful!

Willa

Red line (City tour): Thursday to Monday from 9 am to 4 pm, once every 30-45 minutes. Total duration of 1 hour 45 minutes.
Blue line (Bondi Beach tour): Thursday to Monday from 9:30 am to 3:15 pm, once every hour. Total duration of 1 hour 30 minutes.

We have been to Australia twice for a total of about 6 weeks and NZ for 9 days.

We only did the North island of NZ in 9 days. Australia is a huge country, almost as large as the continental USA.

With only 7-10 days, suggest doing Sydney 4 nights minimum and flying to Cairns or Melbourne for the rest of the trip.
Sydney, you want to visit the Zoo, do a day tour that includes Bondi Beach and the historic places in the city. Also, do a harbor tour, climb the famous bridge piles, and visit amazing museums.
If interested in the snorkeling at the Great Barrier Reef, do Cairns.
Melbourne is great as well, spend 3 nights minimum there and if time to the 2 day tour to the Great Ocean Road and Philip Island to see the amazing Penguins come in from the ocean.

All good advice from the others.
I particularly agree with the suggestion to restrict yourselves to one country or the other and concentrate on that for the very limited time you have available, and since you'll be arriving in Sydney spending time there with maybe a side trip up to Cairns for the GBR would be an obvious choice.
That said, you could get a quick taste of the best of NZ by taking advantage of one of the daily direct flights between Sydney and Queenstown or Christchurch on the South Island. JetStar is the Qantas low cost affiliate and operates (relatively) cheap flights around Australia and over to NZ. Collecting a rental car from one location and driving over to the other over the course of a week would make for a very satisfying introduction to the prettiest part of the prettiest country on earth I think, and would be especially enjoyable if paired with a few days in Sydney - for my money the most beautiful city on the planet.
You're spoiled for choices.

We just got back from a tour of Australia and New Zealand. Honestly, I would spend most of your time in New Zealand. Auckland has a lovely museum and Tirritirri Nature Preserve is an easy day trip from Auckland. Dinner atop the Sky City Hotel was surprisingly good for a revolving restaurant.
If you are Lord of the Rings fans, you must go to Hobbiton. If you go to that area(Rotorua) there are a number of interesting Maori things to see there.
Next to Hobbiton, the best thing about New Zealand, and it's on the other(South) island, is Queenstown. The scenery there is breathtaking.

I would second lovetotravel's suggestions for Australia and also add that I loved the trip from Melbourne to Philip's Island to see the penguins.

You should allow as much time for this trip as possible-there's so much to see in both countries!

Where are you visiting? If you are in the central Sydney area, there is NO need to rent a car. The public transit is excellent. The ferry system is top notch. There are buses out to Bondi Beach and ferries out to Manly or the Zoo. If you want to visit the Blue Mountains and Katoomba, take the train. If you do rent a car, are you prepared to drive on the left side?