If you are visiting as a tourist, you will hopefully not need to be in contact with the medical system. Should you have an accident or get sick, be assured that you want to have a travel insurance. The prices for a doctor's treatment or even worse, a hospital procedure, bears no similarity to what you may expect based on similar services in your home country. Even if you do have insurance, the provider may ask you for a deposit of several hundred dollars that will be refunded once they have been paid by the insurance company. But by that time you will probably be back in your home country and their ability to transfer money into your international credit card is limited... so they will rather send you a check... preferably to an American address. I kid you not. So don't get sick.
For immigrants coming from countries with a public health care system, the American private insurance system can be a bit of a nightmare. If you are on a company transfer visa with travel insurance, make sure that you chose a doctor, dentist, pediatrician and whatever specialists you may need before you actually need them. Many providers don't accept all kinds of insurance and many don't accept new patients. So ask your colleagues or friends if they can refer you, that you may actually have a doctor once you or a loved one get sick.
Medical insurance is generally divided into HMO, managed care, where doctors, hospitals, and specialists are all part of one big network. Kaiser Permanents is an example of such a major provider. The alternative is PPO with more freedom to chose your own doctors but also a higher premium and possibly a higher co-pay.
If you live here, you will naturally also want to have a renter's or home owner's insurance for your belongings, very important is a liability insurance, and a car insurance if you have a car. Note that it is possible in the US to rent a car without insurance. For Americans who have another car insurance with driver's liability, that may work out all right. For tourists, don't skimp on this! Unless you can prove that you have other insurance covering your liability or (in California) $30.000 sitting on a bank account, it is illegal to drive without liability insurance.
The California Insurance Commissioner - or similar institutions in other states - will assist you if an insurance company refuses to cover under their policy. And it really can pay to read the little booklets they send you along with the invoice and understand your rights.
Each state has its own set of rules and the federal government has their taxes on top. In some states there is income tax, in other not. California has one of the highest income taxes in the nation and still highly paid immigrants flock to the state. In Texas there are no income taxes.
In some states there are sales taxes and in others not. Many online retailers have established themselves in these states to save their customers from paying sales tax. But technically the customers would not be abiding by the law if they didn't disclose there purchases and paid the use tax in lieu of sales tax. Lately many such retailers have made arrangements with the rest of the states to claim and pay these sales taxes rather than be barred from operating in some states.
Most states have property taxes but the level varies dramatically. In California the property tax is 1% of the purchase value of the property regulated for change in general market levels if the market goes down but only regulated up with the general inflation if the market value goes above the purchase value. There are special rules if the house is remodeled and there are also special rules for buildings owned by companies - that covers most appartement and office buildings. In some states property tax is the main source of income for the state. It has the advantage compared to income tax, that you can't easily move your property to the other side of the state line. Conversely, in a housing melt down the tax base may erode precariously.
For investment income the tax rates are generally lower than for ordinary income, and for long time capital gains the tax is even lower. The rules were originally intended to encourage investments in risky start up activities with the aim to make new jobs, but they seem over the last decade or two to have become totally unattached to this noble goal.
For companies doing business in many states, this means many tax return filings. And don't expect the IRS in the different states to cooperate with each other.
Social Security and other pensions
One of the many federal taxes employers pay on behalf or their employees is social security tax. This is paid with a certain percentage of the pay, but only up to a maximum amount each year. For high salaries this means that the tax disappears from the payment slip in the last part of the year once the maximum amount is reached. Once you have worked long enough in the US and an accumulated minimum amount has been paid into your social security account, you are entitled to have social security paid out when you reach the age of 65. How much will be paid out depends broadly spoken on how much you have paid in. (With the constant discussion about if Social Security will go bankrupt, there is naturally the risk that what you will be paid many years from now will also depend on the state of US treasury.)
Many countries have an agreement with the US that social security (or what ever the national equivalent is) can be transferred to and from the US. So if you are retiring in the US but have worked for a certain number of years in your home country, you may be eligible for part pension from your home country as well, just as you after a certain number of years working in the US may be able to get social security - even if you have moved back.
Many American organizations have other pension schemes for their employees. For many years it was common to have defined benefit plans. Once an employee had worked for a certain number of years, they were entitled to a pension for the rest of their lives based on seniority and salary. But many of the organizations either went bankrupt - the auto industry and Detroit being examples - or simply can't afford what they promised because people live longer and they didn't set aside enough money when the companies/cities thrived. The government is partly to blame by not having tighter rules on what companies could promise without providing the proper reserves.
These days most retirement benefits are through so called defined contribution plans, and many companies encourage employees to save for retirement from their paychecks by some matching of the savings. These retirement accounts are often known as 401(k) plans from their number in the tax code or IRA, Roth IRA and others exotic names (having nothing to do with the Irish Republican Army but all things American must have an acronym.) Be aware, that many companies prefer to match with own shares which may leave your financial well being too much exposed to the ups and downs of one single company. Unfortunately many employees don't take advantage of these offers and leave money on the table every year that they would surely have been glad to have later in life.
Other boring things you should know
The US legal system is based on case law where most of continental Europe is based on what is known as Roman Law. The difference is that Roman Law typically creates a common foundation from which parties can't make agreements that are too skewed to favor one party. In Case Law, a contract is a contract unless through a judge or in arbitrage it is decided that it was not reasonable. But that requires that you take it to a lawyer or before the court. So you have to read the contracts before signing at the X and if you also understand what they say, so much the better.
An example is that you promise not to sue somebody who provides a service to you - regardless what that person does. You take your kid riding? Sign a liability waiver so broad that if an employee at the establishment runs down your kid with a tractor while intoxicated, they go scott free. Go skiing, and nothing the groomers and lift personnel may do that will harm you can ever be their fault. There are two responses to this. First, if only one parent signs waivers the other can sue, should intoxicated tractor drivers come bouncing. Second, not all contracts hold up in court. But it illustrates how America has become a litigious society.