Walking distance to a train station. Even if you’re in the inner suburbs. If you can get to a train, you can go anywhere.
Well, anywhere that also has a train station or a decent bus service.
The obvious answer is inner city. The overwhelming majority of public transport in Brisbane is based on taking people in and out of the city. If you're already in/near the city, then anywhere you want to go is probably only going to take one bus or train; two at most.
I lived in Yeronga without a car for 10 years. I hardly ever used public transport either (mostly because it was so inconvenient it was a whole day trip if you ever wanted to go anywhere). But I could do it because I was comfortable walking a lot. It was a 10 minute walk each way to get groceries, a 45 minute walk (4 km) to where I worked (so no gym membership needed), and being an introvert I didn't enjoy going out on weekends so that made it easy. The walk home from work was past the shops so I'd just buy a little (at least as much as I could carry) every 2-3 days on the way home from work, so I rarely had to make a specific trip to the shops.
If you're an extrovert and you have to be out in public all the time, the other advice here about living near good public transport connections is wise, but if you're an introvert it's probably beneficial to live within walking distance (or at least a short public transport trip) from where you work, and not too far from some grocery shops. That way food and work is taken care of.
I will admit though, after finally getting a car it's much nicer to be able to pop down to Bunnings to get something and be home again 30 minutes later, instead of having to set aside a whole day for the journey, or order it online and have to wait a week or two for it to arrive. I guess if you can afford it then a taxi or Uber could be another option, but if you do that too much then it won't take long before it's cheaper to own a car.
I ended up buying a cheap ~8 year old ex-taxi hybrid, which only gets driven once or twice a week so uses hardly any fuel (around $200/year) so the overall cost ended up being much less than I expected. It has made me a lot lazier though as I will drive the two minutes to the shops instead of walking for 10, so with WFH also reducing the amount of walking I'm doing I have put on quite a bit of weight (20 kg) so going without a car and having to do so much walking was certainly much better for my physical health.
In addition to trains, I'd suggest looking at being close to major bus routes. Unfortunately, the practical reality right now is that Brisbane is more of a bus-based city than a train-based one, and so ruling out buses when considering where to live limits you dramatically.
But don't just look at if there's any bus nearby. Look for one of the major "BUZ" routes. Or even better, one of the new Metro routes (or potentially the proposed future Metro routes). Anywhere on the South-Eastern Busway is likely to be good, as is the Northern Busway. To the North, I'd say anywhere as far out as Chermside (but not much further) would be viable if you're within a good distance of a stop used by the 330, 333, or 340. Beware the 370 though, which is a much less frequent and slower service.
Consider your needs outside of just commuting to the city. Where is your social life? Will you be able to get to friends' houses, sporting clubs, and other social activities you enjoy from where you live, at the times you'll likely be doing it? Are there shops within walking or safe cycling distance? These are usually the hardest problems to solve.
