Living between Melbourne and London I've used Qantas for intercontinental flights many times. I've also flown from Melbourne to New York via Los Angeles once a few years ago. Qantas is as comfortable and reliable as any other airline that I've used to fly ultra long haul and prices are competitive. I have never had one of my Qantas flights delayed. It has a reputation as being one of the safest airlines in the world. Cabin crew are friendly and helpful, but take no nonsense. Fellow passengers are cheerful and well-behaved (I've been on other airlines where this hasn't been the case). Connections between Australian cities are excellent and you also have the option to book an LA ticket via Brisbane or Melbourne you're planning to visit more that one Australian city.
I'm not sure which type of aircraft Qantas currently uses to fly between LA and Sydney, but I'd strongly recommend booking on an A380 if that is an option. In economy and premium economy Qantas A380s are by far the most comfortable long haul planes I have flown on (and I've been on 747's, A330's, 777's and 787's to be able to compare them).
Qantas is one of the first airlines that I consider when booking ultra long haul, but I will choose others if prices are more competitive or connections between destinations are more efficient. For example, I've just booked Singapore Airlines for a Melbourne - Paris trip because of much better connections.
If you're flying from LA to Sydney you'll have a direct flight, which you're likely to find more comfortable than using Asian hubs like Singapore. I wouldn't fly USA-Australia via the Middle East - you're going in the wrong direction and have an unnecessarily long journey. Be aware if you're in a premium cabin and flying out of Melbourne International terminal, the furniture in the Business Class lounge was looking shabby the last time I used it about a year ago and I've subsequently had others tell me that it hasn't been improved. Also be cautious that any internal Australian flights you book are not Jetstar (Qantas' low cost subsidiary) code share tickets.
Regardless, enjoy your honeymoon! Australians are friendly, the food is great and the weather is (mostly) good!
SLFAussie