How do I get my first Social Security number or an ITIN when I arrive in the US?

Which one you need depends on work authorization. If your visa lets you work (H-1B, L-1, certain F-1/OPT), you're eligible for a Social Security number, apply at a Social Security office with your passport, visa, I-94, and employment documents; it's free. If you can't work but still have a U.S. tax filing obligation (a dependent spouse, an investor, a landlord), you instead apply for an ITIN (Individual Taxpayer Identification Number) using Form W-7, usually filed with your tax return. You can't have both for the same person, and you should never use an ITIN once you become SSN-eligible. The SSN is also what banks and lenders use to build your credit, so get it promptly once eligible.