Keep in mind that several of the people giving advice here may not have ever made an overseas move from or to the U.S. You are going to encounter some things that are just uniquely Immigrant Stuff, and sometimes that will be super annoying. As such, look for a good immigrant community to be a part of (online, offline, or both). If you can find one that's specifically for French immigrants, it will be even better for you, as no one will have a better idea about what might be surprising to a French expat than other French expats. :)
Here's my more finance-related advice:
Learn a little about U.S. banks before you move. Many of our big banks provide an adequate service, but not necessarily the best; they're known to have tons of fees and offer little in return. Some people will advocate credit unions, which are smaller, (usually) locally owned banks that are focused on certain communities or groups of people; they charge no fees. Personally, I would recommend you use Ally, which is an online-only bank, with the only drawback of not being able to deposit cash, which isn't likely to be a big concern; you can withdraw money from any ATM, though, I should note, and they'll reimburse the ATM fee at the end of the month. Ally gives better savings account interest rates, and they have amazing customer service via online chat or phone, so if you were ever uncertain about something regarding your banking, they'd be really good for you to deal with. They also have a good phone app.
If you need to transfer any money to/from France, use XE. Do not use your bank or a traveler's currency exchange system, as they're both rip-offs. They take huge cuts for themselves. XE doesn't work that way.
Since you will be employed by someone, your taxes are going to be easy. Don't fret. You can use tools like TaxAct or TurboTax to help guide you when the time comes, if you find the forms too daunting. Do note that how much tax you will pay will vary considerably according to which state you're in. Everyone pays federal income tax. Most states also have a state income tax. Same goes for other taxes, like the sales tax — some states have no sales tax, low sales tax, high sales tax.
It's almost certain that your employer will provide you with health benefits (whether that includes vision and dental, I don't know, but basic health coverage, yes, most likely). Make sure you understand your benefits! I cannot stress this enough since you're coming from a place with universal care. We provide great care here, in my experience, but the payment system is completely fucked, and you are in for a huge culture shock when it comes to this, even if you know the stories. YouTube is actually a surprisingly great resource for understanding some of the complex insurance concepts. Here is a video that explains deductibles, copays, coinsurance, and out-of-pocket maximums. (There are lots of other similar videos if that one isn't helpful.) Here's one that explains different kinds of plans, including HMO, PPO, and HDHP.
If you decide at some point that you need a financial advisor, I'd go by online reviews if you're in a city, unless someone you know and trust can recommend a particular person. One of the biggest advantages (and sometimes one of the biggest disadvantages) to living in the U.S. is our large population. There is just a ton of data for everything that really removes a lot of guesswork and risk-taking. Need a doctor? Look at user reviews. Need a financial advisor? Reviews. Want to go to a good restaurant, but don't know of any? Reviews. Yelp, Foursquare, Facebook, and, to a lesser degree, Google all have tons of reviews for tons of places. I would imagine you already use some or all of those services in the same way in France, but there's no doubt going to be more content here in many cases, just because of our numbers. It's really helpful, and very rarely has the hive mind let me down.
Without a U.S. credit history, the first complex you rent an apartment from will likely want to know a little extra about your employment and take-home pay and/or they'll want to see a bank statement that proves you have some extra money in the bank. The bank statement will not have to be from a U.S. bank. All they care about is the money, the "Can you pay?" If the answer's yes, you won't run into any trouble.
If/when it comes time for you to buy a phone here or enter a phone contract, I'm going to strongly suggest you go with AT&T or T-Mobile, as they offer widespread coverage and use GSM (meaning the phones have removable/replaceable SIM cards). If you don't do this, you won't be able to use your phone when you visit home just by switching out the SIM card. Verizon and Sprint, the other two big carriers here, use CDMA, which locks the phone to the carrier (i.e., there's no SIM).
It's good you're trying to prep now, but there's only so much you're going to be able to do while in France. You will not be able to prepare for everything, and that's okay.
I hope you enjoy your time here! If you're headed to Seattle, drop me a line!