Generally, in the US, the amounts are calculated annually. So, 4.2K a month would be $50K/year. For a software engineer in California this is extremely low. You can check the salaries on various comparison sites, for example - glassdoor.com. Choose the area, similar companies and job descriptions and see how much people earn. With several years of experience, a Software Engineer in California would earn at least double that amount.
The reason is that California is a very expensive State to live in. It is the State with the highest income tax in the US (except maybe for New York, and I'm not even sure about that), and the highest cost of living. Even in the areas like Los Angeles, or even Central California, it is much more expensive than most of the US.
Back to taxes - you will probably be considered as self-employed. So you will pay, for starters, 15% self-employment tax (equivalent of what you and your employer would pay in FICA taxes had you been employed, a US equivalent of National/Social Insurance systems you're familiar with in the EU). After that, depending on how large your family is and how much of other income you may have, you'll probably not pay much in taxes - maybe another 10-15% in total. So assuming you paid 30% in taxes, you'll end up with $35K a year net income, which is $3K a month. You can live with that, but depending on your life style - it may be very tough. In the Silicon Valley - a decent 1bd apartment will cost you between $1.5K and $3K a month just for rent.
Keep in mind that you'll have to buy your own health insurance policy, which will cost you between $500 and $1500 a month, depending on the coverage options (under the ObamaCare exchanges). It may be tax deductible if you're self-employed, but it is a significant expense.
The solution of continuing working for your Italian company is excellent in the short term for you to move and acclimatize in California. But you will be looking for a job very quickly.