It will not solve itself eventually. I have a friend who immigrated to the US and went this route, and he basically lost a few years he could have used to build a credit history, which hurt him ten years later when he wanted to buy a house.
So when I immigrated to the US, I started with a secured credit card as fast as possible. Mine was free to have, and you could set it up so it was paid automatically, in full, from your checking account every month. That way you just need to charge something to it each month, and you'll slowly build your history.
> you just need to charge something to it each month
You don't even have to do that. You can leave it idle for months if you want. The only thing you have to watch out for is the issuer canceling inactive cards. Having an idle credit card shows you aren't maxing out your cards - you have credit available that you are not using. Length of credit history is also a factor in your score.
So when I immigrated to the US, I started with a secured credit card as fast as possible. Mine was free to have, and you could set it up so it was paid automatically, in full, from your checking account every month. That way you just need to charge something to it each month, and you'll slowly build your history.