First, get the SPG Amex and make it your primary card. That's the easiest decision to make. That will yield free hotel stays fairly quickly, or you can convert the points to miles (although I would recommend doing it only as a last resort to cover an immediate award need). The only drawback to this card is that it is not a good option for augmenting a UA account.

Then, I'd pick one of AA and UA. Their programs are roughly similar, and they have different partners going to Asia (Star Alliance vs. OneWorld). Based on schedule and average costs, pick the one that suits you best. If I were in LAX flying the routes you mention, I think I'd pick AA. But some people here might have a good UA argument as well. Neither of those two is a bad choice.

Finally, get the associated MC/Visa credit card for that program. For small-timers like me, I'd say skip this step - I don't get enough non-Amex overflow to justify the fees, so I just use my Marriott Visa even though it's not an optimal rewards card for spending outside of Marriott hotels. But if you're running $100k/yr. in expenses, you'll find that a good chunk fall through to MC/Visa - enough to justify carrying the airline credit card and picking up the auxiliary perks that come with it. The UA card, for example, comes with some upgrade and flight discount certs. The AA card comes with some discounted awards and is occasionally tied in to a nice mileage-bonus promo with AA or a hotel chain. Perks that in and of themselves aren't of major value, but are enough to nudge you in that direction as opposed to a no-annual-fee CC with near-1% cashback.