After the US Congress passes a new bill and the president signs it into law, when does this law come into force (become effective and binding)? Does the law need to be oficially published somewhere?
By default a new US Federal law is effective as soon as it is enacted. That is, when the President signs it, when the 10 days for approval expire without the President vetoing it, or on the date when the second of the houses of Congress votes to override a veto.
However, in modern usage, many, perhaps most, statutes specify as pone of their provisions an effective date some time after the date of enactment, in some cases several years later. If such a date is specified, that is the legal date.
All bills and the actions enacting them are recorded in the Congressional Record, which is a public document, published on a regular basis. They are generally included in the next revision of the USC (united States Code) and the USCA (United States Code, Annotated). But these publications are generally well after the date of enactment.
The Thomson Reuters Perspectives: Teaching Legal Research and Writing (winter 2000) “HOW Can I Tell the Effective Date of a Federal Statute?” by Barbara Bintliff says:
The generally applicable rule is easy to state: a federal statute usually takes effect on the date of its enactment,3 which is itself determined by when the last step necessary for the statute’s enactment occurred. Most of the time the last step in the enactment process is the president’s signature, and this date, therefore, becomes the date of effectiveness for most federal statutes. However, there are exceptions to the general rule. For instance, sometimes the last step occurs when Congress overrides the president’s veto, or when the bill is allowed to become law without a presidential signature. And sometimes a later date is specified by the terms of the statute itself.
The article geos on to say:
The date of enactment is most easily located by consulting the parenthetical information provided after every statutory section included in the print and online versions of the United States Code (USC), the United States Code Annotated® (USCA®), and the United States Code Service (USCS).5 The parenthetical is variously called a “source note,” “statutory authority,” “statutory history,” “history,” “source credit,” or “credit.”
In the print versions, the parenthetical follows the statutory section with nothing in between the statutory text and the source note. In the primary online versions, it is labeled “credit” (in USCA on Westlaw) or “history” (in USCS on LEXIS-NEXIS). The credit has been included with federal statutes since they were first codified in the Revised Statutes of 1874 and is an official part of the statute.
"Enactment of a Law - Learn About the Legislative Process" from Congress.gov describes the procsss also, saying:
If the president does approve it, he signs the bill, giving the date, and transmits this information by messenger to the Senate or the House, as the case might be. In the case of revenue and tariff bills, the hour of approval is usually indicated. The enrolled bill is delivered to the archivist of the U.S., who designates it as a public or private law, depending upon its purpose, and gives it a number. Public and private laws are numbered separately and serially. An official copy is sent to Government Printing Office to be used in making the so-called slip law print.
If a bill which has been vetoed is passed upon reconsideration by the first House by the required two-thirds vote, an endorsement to this effect is made on the back of the bill, and it is then transmitted, together with the accompanying message, to the second House for its action thereon. If likewise reconsidered and passed by that body, a similar endorsement is made thereon. The bill, which has thereby been enacted into law, is not again presented to the president, but is delivered to the Administrator of the General Services Administration for deposit in the Archives, and is printed, together with the attestations of the secretary of the Senate and the clerk of the House of its passage over the president's veto.