Hello and welcome. I'm a US citizen, but I also moved from abroad to the Bay Area for work, so I have somewhat similar experiences. Sounds like you have a computer job that pays decent wages, but the Bay Area is expensive and has a lot of issues around housing availability. Finding housing may be a problem, even at that price range.
When I worked in Santa Clara, I lived in downtown San Jose because I was more used to living in a city. But San Jose isn't much of a city, and all of Silicon Valley is more of a suburban sprawl. So be aware that it is the suburbs if you come from the city. The apartment complexes are boring in my opinion, so I found a room in an old Victorian house--but you won't have that choice in Sunnyvale. Sunnyvale does have a little downtown area, more of a main street with more dense apartments now, so maybe that is an option. Having an efficient car to get out of town and visit the mountains on weekends is a great idea, I used to have a civic because they were only about 5k back when priuses were new.
I'm glad you are considering biking, not just because it's the mustachian thing to do, but there are too many cars already and it's depressing to drive in silicon valley. I did bike 6 miles (10km) each way to work, which is about the limit of practicality around 30 minutes. It only freezes a few times in winter, there isn't much rain, but it does get hot 6 months of the year. Biking in Silicon Valley is a mixed bag: it's not crazy like some cities in Europe, but it doesn't have many bike lanes either--or they just paint a lane on a big road and cars cross it all the time. It's not full of trucks that want to kill you like Texas, but people are not very aware of bicyclists--always pay attention. There are lots of large roads made for cars and lots of traffic, which makes them hard to cross. There were literally some intersections that I would enter on green and not be able to cross before red. For the weekends, there are some nice bike trails by the bay and along the creeks.
I would look for apartments that are close to cultural things and shopping for your wife. Like the downtown area, the library, and a park (also good for little kids). That way she doesn't feel stuck at home with driving as the only option.
That's great advice about the credit card, just be sure to use it a bit (groceries, etc) and pay it off every month--treat it like a bank card, not a line of credit.
For healthcare, companies usually provide good coverage with the choice of PPO (select your own doctor) or HMO (a company that manages hospitals and doctor offices, usually Kaiser). Child birth should be covered, but it's always good advice to read all the paperwork and choose the one that's better for you (PPO usually costs more than HMO--the part you pay every month through work, and the copay when you visit a doctor). Companies also offer life insurance and it costs extra to cover your wife, but I personally choose to spend a few extra dollars on that every month--especially with kids myself.
For the 401k, it's great if you can max it out, but you may need a little more cash in your first months, so you could start out with the minimum to get the employer matching contribution, if they have one. You can always increase it later at any time. Another thing to consider is that 401k is retirement money, so it's locked away for a long time. Until your situation is more certain, you may want to save money in a regular investment account (like vanguard). It will be taxed, but it will also be easy to move the money if you need it. You will need to open the account yourself and set up a monthly transfer.
Speaking of money, you will need to open a bank account. You have the choice between big corporate banks (Bank of America, Wells Fargo, Chase) and coop banks (sometimes called mutuals in Europe). I have used both, the big banks have more ATMs to get cash, the coops are more ethical and community oriented. In any case, a credit card is good to have because you can use at to pay almost everywhere now.
Silicon Valley is a big mix of cultures, but they're not always celebrated out in the open. If you didn't go look for it, you wouldn't find the Latino, Korean, or Chinese communities and events. So be open to exploring the area and finding other things to do, not just computer culture and suburban living. I know you'll also hang out with other Russian immigrants, it's only natural, but also try to meet other people.
This is true for food as well--you don't have to shop at the bit American store chains (like Safeway), there are lots of Smaller choices and you can avoid American processed foods. For example, I used to buy sandwiches at the Vietnamese shop near my appartment in San Jose, and I'd buy most of my vegetables at a small store run by a lab ease family. There are farmer's markets, but only once a week and you have to find them--and they tend to more expensive but better quality.
Of course, there is always San Francisco, which has a lot of nice things to see and do, and where culture is more open, but it's almost one hour away with frequent traffic jams--but you can get there on the train.
If you like the outdoors, there are lots of hiking trails in parks and open spaces around the Bay Area (it's a great place for that). For example you can drive less than an hour and hike in redwood forests. Many places are free, but state parks charge a reasonable small fee--or you can get an annual pass. Just be sure to learn about poison oak, so you can recognize it and avoid it. And don't pick the mushrooms because they are different than in Europe (join a club and learn to recognize local mushrooms if you're interested).
I know this is too long, but I remember how it was an exciting but uncertain time for me. I wish you he best of luck, and feel free to ask more questions. There must be a lot of Bay Area mustachian on here.