I'm moving to the US this year. When can I start making contributions to a 401(k)? Can I start as soon as I get my first pay check or do I need to wait until I file my taxes?
401(k) is a payroll deduction, so you can start whenever your employer says you can start. Some allow starting (almost) immediately, others will have you wait till the next year. Not every employer provides this benefit at all.
Do talk with a tax adviser who can tell you how it would affect your tax liabilities in your home country before doing anything though.