Fiji with 23 month old twins. Which Island(s)
 

We will be traveling to Fiji before making our way to Australia and New Zealand. What 2 islands should we stay at? Ideally, I would like my 2nd island to be easier to get to the airport when jetting off to Australia to avoid stress making it on time to yhe airport. We are flexible on the amount of time we spend in Fiji, but would like to limit to 12 days or nights.

Get the transfer hassles over first and head to one of the 'real islands' first. I'd pick a hotel at Denarau for your second stay as it's closer to the airport. Also you can walk to the other hotels and resorts and use their pools and restaurants.
My niece just took her family (youngest 3) to Treasure Island and they had a fantastic time. My friend and her family with 3 year old granddaughter went to Plantation and it was just as great.
We have been to Denarau (on the main island of Viti Levu) which was lovely and the airport is about 20 minutes away. We knew it was not the 'island experience' as such but we just needed a relaxing break away and it was fine.
How well do the toddlers travel? There's a lot of travelling involved with Fiji, New Zealand and Australia. We are all about 3 hours flight apart so the travel will take up most of the day.