stueandmel, From my experience the best bet is to checkin as early as possible. I'm not sure whether there is any priority order for getting the seats.
We did a number of trips with our daughter from when she was about 5 weeks, (Syd - Mel) and also to the UK when she was 2.5 months old (plus a number other domestic).
For domestic the answer was always 'request it at checkin'. For international we had seats pre-allocated and had the bassinet but we were travelling business class. However, A women on the same UK flight as us in business class was told they couldn't reserve a seat with a bassinet for her. Not sure why she was told that. I spent about 10 mins on the phone with QFF while they tried to get seats with bassinets for us.
Coming back from the UK, we could only get one of the seats with the bassinet and one directly behind. Fortunately the person seated with the bassinet had no baby and was happy to swap seats with me so my wife and I were seated together with the bassinet.
The last time we requested the bassinet on a Syd - Bne flight at checkin, we were told we had the seats with the bassinet but in fact we didn't. We were 2 seats back. Because it was such a short flight, we didn't bother asking the other person to change seats with us.
If you can't get the seat with a bassinet, just make sure you get a few pillows from the FA before takeoff.
And also request an infant seat belt as soon as you get on board. It goes around the baby and then attaches to your belt.
My wife used to feed our daughter during takeoff because that would often put her to sleep and we'd heard that it could also help her ears equalise. On some flights my daughter has slept the entire trip!
we also tried to organise flights around my daughters regular sleep cycle. This sometimes worked

and other times didn't
I think we only had one bad trip with her where she cried for about 20 minutes before landing and promptly fell asleep just as we were landing.
My take has always been that everyone is entitled to travel by air including children and babies. You do what you can to ensure they don't disturb other passengers comfort but if they cry and scream and you've tried everything you can to calm them down, that's all you can do.
Have a good first trip!