Australia to west & east coast US: which order is better?

Hilmar basically has it, but let me just add this…

Definitely agree on West is much easier than East.
I spent a decade travelling from the UK to Japan, sometimes for as much as 6 weeks at a time. It was a 24h journey, starting about 5am, arriving around 4pm [local times]; meaning I had to try stay awake another 6 hours or so before first groundside sleep.
Because of flight schedules for all the others I would be meeting, they would always arrive late so I had to entertain myself for that first evening, or try my best. I never made it past 10pm.

I tested many variants on this over the years, the 'staying awake first day' is definitely the best approach.

The rest comes later.

In the first few days you get better at staying up 'late' but at the same time you start to wake early. This lasts about a week…
Then they eventually overlap.
You can't get to sleep at night & can't wake up in the morning. You're tired all day & awake all night. It takes about another week to get over this. Then you're fine.

Coming home - back West. No problem at all.

Conversely, I did several equivalent trips the opposite way, London to LA. Not an issue at all. I paid when I came back home.
I also had a decade to compare notes with my colleagues. Going to Japan, the Europeans all had this issue. The Americans all suffered after they got home.

I would therefore recommend, as you're going to suffer this whatever you do, that you take your punishment early & visit New York before San Fransisco. If you suffer a bit in New York, you'll gain at least a little back in San Fransisco.*

Notes: when you're on a marathon journey, time stops… or lasts forever [one or the other, or they're both the same. It becomes hard to tell.] Your internal clock seems to ignore any sleep you got on the plane, once you arrive.
You will never feel 'normal' on your first day there & will suffer this 'revolving clock' as your going to bed & waking up times cycle past each other.

In summary -
East is hard, West is easy.

*There's another potential factor in this, which I've never had to contend with myself, having never done both cities in one jaunt. In Summer, NY will probably not get dark until 9 or 10 at night. SF will be dark by maybe 7 or 8. This is going to affect your perception. The further South you go in the Northern hemisphere, the closer to perpetual 12 hours daylight you get. For someone like me, used to Northern climes, this makes it feel like the sun just fell over the horizon like a dunked basketball, even in the middle of Summer.

That depends a bit on what flights are available at your specific dates and budget and what you personal pain points are.

A few things to consider.

1. In my experience the best way to deal with jetlag is to immediately hard-switch to the new time zone. Don't take naps during the day, don't go to bed early, don't think about "what's the time in the other place".
2. With respect to 1): early evening, late afternoon arrivals are the best for long hauls. Go to the hotel, take a shower, have a light dinner and a quick walkaround, collapse into bed around 10pm.
3. Early morning arrivals are the worst: you are typically sleep deprived but still have a full day to go and often can't check into the hotel yet. A day hotel can help with that. Super late arrivals (past midnight) can also be difficult but depends a bit on what happens the next day.
4. Early morning departures are also not great if you have to get up at 3am in the morning.
5. Connecting flights vs. non-stop has pros and cons. Some people prefer breaking it up, other like to get it over with in one go and/or maximize the length of the longest leg. Connecting flights also tend to be cheaper (because of illogical airline pricing).
6. Personally I find going west easier than going east. I've done east<->west coast a lot and hardly notice going west. Coming back east is more difficult especially if you taking a red eye.
7. Stating the obvious: flying in business makes a huge difference here (both in experience and in price).

So in summary, I personally find that optimizing arrival and departure times can really help here. I found going to China (12 hour time difference) with "good" times is easier for me than the stupid red eye to Europe, despite being only 6 hours time difference.