I am familar with the issues of new immigrants that lack an SSN - not least because I was one myself less than four years ago.
My advice is as follows:
(i) be patient, getting a credit history takes time.
(ii) you WILL have to pay more fees/ downpayments, which WILL make life more expensive for you in the short term. This includes renting a home, probably from a smaller, private, landlord as they are more likely to take the time to find out whther they want to rent to you rather than relying on a credit report.
(iii) realistically, you may not be able to afford a car for a month or two, until you can get together the cash to buy outright an old wreck - then you can trade up later when you have more cash and access to credit. I guess that until then you'll have to make do with public transportation and taxis.
(iv) avoid making applications for credit (cards) that you are unlikely to qualify for, every rejected credit application is another negative mark on your credit history.
(v) utility services file credit reports, so get utility bills in your name as soon as possible (you are almost certainly going to have to pay an up-front deposit to get connected - see (ii), above).
(vi) avoid getting a "secured" credit card if you can possibly manage without - many credit providers look badly at owners of "secured" credit cards as they are typically used by US residents who have major credit problems e.g. bankruptcy, or a history of credit defaults and write-offs against them.
(vii) be patient, getting a credit history takes time.
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This is my opinion; it is not legal advice, nor the view of my employer, and it may change tomorrow.