What requirements should I meet to start a bank in USA?
What are the requirements in other countries?
For this answer in general terms, a "bank" in the US is either a commercial, for-profit bank, or a non-profit credit union. (There are many types of financial institutions that are under federal regulation: investment banks, commercial banks, brokerage houses, insurance companies, etc.)
There are full outlines at The Federal Reserve on the process to open a bank, who may open a bank, and what approvals are needed. All banks in the US fall under Federal Law, though there will be state laws that apply, too.
Starting a bank involves a long organization process that could take a year or more, and permission from at least two regulatory authorities. Extensive information about the organizer(s), the business plan, senior management team, finances, capital adequacy, risk management infrastructure, and other relevant factors must be provided to the appropriate authorities.
The proposed bank must first receive approval for a federal or state charter. The Office of the Comptroller of the Currency (OCC) has exclusive authority to issue a federal or "national bank" charter, while any state (and the District of Columbia, Guam, Puerto Rico, and the Virgin Islands) may issue a state charter. Before granting a charter, the OCC or state must be able to determine that the applicant bank has a reasonable chance for success and will operate in a safe and sound manner. Next, the proposed bank must obtain approval for deposit insurance from the Federal Deposit Insurance Corporation (FDIC). Additional approvals are required from the Federal Reserve if, at formation, a company would control the new bank and/or a state-chartered bank would become a member of the Federal Reserve.
All insured banks must comply with the capital adequacy guidelines of their primary federal regulator (Federal Reserve, FDIC, or OCC). The guidelines require a bank to demonstrate that it will have enough capital to support its risk profile, operations, and future growth even in the event of unexpected losses. Newly established banks are generally subject to additional criteria that remain in place until the bank's operations become well-established and profitable.
For Credit Unions, the National Credit Union Association responsible for the chartering of federal credit unions, which are nonprofit and cooperative in nature. https://www.ncua.gov/services/Pages/field-of-membership-chartering.aspx
The Federal Credit Union Act (of June 26, 1934) enabled credit unions to be organized throughout the United States under charters approved by the federal government. The purpose was to make credit available to more Americans and promote thrift through a national system of nonprofit, cooperative credit unions. The Federal Credit Union Act also is the source of authority for all federally chartered credit unions and governs the coverage and terms of insured accounts at all federally insured credit unions. It also determines the structure and duties of NCUA.
The Federal Credit Union Act: https://www.ncua.gov/Legal/Documents/fcu_act.pdf
For the rest of the world, Google is your best bet. All countries (hopefully) have extensive regulations governing their banking industry, who may open a bank, and how those banks work - whether public, private or government owned - with the banking industry in the rest of the world.