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.

I have flown between Australian cities including Sydney, Melbourne, Cairns, Brisbane, Canberra on multiple Australian trips. Used those cities as a base for further sightseeing. Depending on your touring interests, a car is not necessary.

From the cities I named above, I was able to easily find very comfortable day trips, shuttles, and Ubers to sites I wanted to see. In fact, I became spoiled by the ease of touring Australia sans automobile. I now am irritated that I don't find the same availability in other countries.

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.

Just saw "New Zealand" in your heading.

I have also had a trip to New Zealand. Once again, I choose not to rent a car - there is quite good bus service between Auckland and Rotorua (for instance) and I flew between Rotorua and Wellington. (I have not visited the South Island.) Using those cities as a base, I found shuttle services, day trips, and Uber as well as public transport to be sufficient.

I think your best approach would be to find a guide book on Australia and/or New Zealand and see what attracts your interest. There are also a number of posts within the past year on traveling to both countries on this board. The search function isn't great, but check within the "Beyond Europe" category and (I think) you can filter by date.

With only 10 days, I, personally, would pick either Australia or New Zealand and limit sightseeing to no more than 3 places within that country. (3 places may be pushing the itinerary, but I like to see more "in depth" than some - you will know your preferences. Good advice above from Geovagriffith.)

My guess is that you can find ways to avoid renting a car if you wish to do so. If your interests are slightly more remote areas, this may be less possible. For example, I researched Tasmania and Kangaroo Island and discarded the notions for myself as too limited without my own car. I believe there is some advice on RS posts if these areas appeal to you.

I have flown via Hong Kong and Singapore (expat at the time) to Australia; direct from Chicago to New Zealand. My brother and sister-in-law were delighted to find the best route home to MidWest US through Bali!! Lots of options if you use a creative approach.

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.

A pre-emptive welcome to Australia.
You included NZ in their but haven't mentioned how that fits into your plans.
What sort of things do you want to see and what sort of experience of Australia are you looking for?
Just a heads up my country is huge, 12 times the size of Texas but has about the same population.
Nearly half of our population live in the Metropolitan areas of Melbourne and Sydney. So quite a contrast between the big cities and the other regions. Where I live the population is 200 (but also many 1000s of kangaroos) and the nearest shops are 15 kms away.

Do you need a car?
Seeing you are doing such a short trip then not really.
In my state of Victoria you can reach most destinations of interest by train or bus or often a combo of the two.
Within the metro areas public transport is extensive.
Jetstar, Qantas or Virgin fly interstate between the bigger cities and a few smaller ones popular with tourists.
If you give me an idea of places of interest for you I can probably provide some specific info.

Yep long haul flights aren't much fun particularly in the cheap seats. When we fly to Europe we usually spend at least 25 hours each way in plane seats. The whole journey from door to door is usually about 35 hours each way.
Never been to the US so know nothing about those flights.
Last couple Euro trips we have utilised the Transit hotel at our first stop at Changi (Singapore). We have booked either 4 to 6 hour stays. Just a chance to lay down and sleep. Not sure where your trip would be broken but check to see if their are Transit hotels offering this service on your route.
Cheers

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?