Starting your US credit journey really depends on how much income/cash you have. Unfortunately, as with everything in life, money allows more freedom in getting things done that help.
The thing that would have helped me the most was to have an AMEX card before arriving in the US. You could then use the global transfer program (as you guys have done) and get a US AMEX issued instantly. Unfortunately, AMEX cards are pretty much useless anywhere else so I closed mine a long time ago. That would have saved me time and money. Oh well.
If you can't do the AMEX thing, opening a secured CC at a credit union is the next thing you should do. Check out SDFCU or your local CU. My local CU had a great secured product - Visa platinum, no FTF, no fees, really low APR and a decent 1% cash back program. Put as much money as you can afford to lock away into securing the card (the optimal amount is $5,000. Locking that away is a bit rough on anybody, so try something like $3,000 if you can). Some CUs will open a secured CC without an INQ.
Once you have your card, PIF every month for 6 months. Right before the sixth month gets reported, do whatever is required to get it to report a util of 1%. You'll now get a score on Credit Sesame. When a score shows up in CS, you now have a FICO generated (though your CS score isn't a FICO score, blah blah blah). At this point, a clean file with 6 months of history and 1 TL with a CL of 3000-5000 and 1% util will net you a EX FICO of 750. Congratulations, you now qualify for any card on the market, except for one small problem. You only have 6 months of credit history.
There's been some talk of Chase issuing Freedom cards at 8 months, but they will definitely not issue a CSP or Ink at 6 months. You need to wait until at least 12 months of history have reported. Your only real prime option at this stage is AMEX.
With a high enough income and an EX FICO of 750, AMEX will issue you a charge card (the PRG is particularly good for the first year). Once that happens (and you can actually do this a minute after you get approved for the charge card), you can apply for a revolver like the SPG (also no AF for the first year). AMEX will collapse both INQs into one for both cards. Your CL won't be great (roughly the same as the highest CL you have) and your APR won't be great, but it's a start as your first unsecured CC.
Wait 61 days, do the 3x CLI. You now have over $10K on a TL. You're doing better than most of the US population (if you measure doing better by CLs...). Now, you can wait until the next AMEX CLI (6 months from last CLI) or apply for the CSP right after 1 year of history. You will be approved (probably not instant, the screen will ask you to call in and reference a number -- you're approved, this is just fraud prevention). Your CL on your CSP will match your largest AMEX CL + ~$2k. I don't really know what else you need from this point on, but you should try not to waste INQs until your AAoA has built up. My experience is that income > AAoA > INQs, though.
Whatever happens after that is up to you and your income level. Of course, YMMV with all of the above.