Maybe one of our resident sharks could weigh in here.
Well, I'll be glad to try to explain how the process works although I'm a little uncertain of what question I'm supposed to be answering. Everybody here is at least partially wrong (Jerome, of course, the most egregiously). I'll just make up my own questions and answer them, I guess.
Can a President sign a bill and then claim it's unconstitutional?
Yes, although he probably wouldn't because it's insane. More likely, the President would claim that only some certain application of the law is constitutional and refuse to apply it in other instances. For example, Connecticut passed a law banning the use of contraceptives. The governor (through his attorney general), refused to apply the law to married individuals because, he conceded, they had the right to use contraception if they wanted.
Can the legislature sue the President for refusing to enforce a law?
Not usually. Only an agrieved party can sue anybody for anything. Chances are that the legislature is not the entity hurt by refusal to enforce a law. The soldier who doesn't get his pay raise because the President won't release funds that Congress has authorized could sue; the Congress could not. Congress could sue the President to enforce a law in which they were harmed such as if the President failed to turn over recordings of him agreeing to a bribe.
Can a regular person really sue the President?
Not usually. One generally sues the person responsible for administering the law. The soldier waiting for a pay raise would probably sue the Comptroler of the Army or the Secretary of Defense. In any case, the President would at most be a defendant. He would never be a plaintiff in any suit to enforce a law (by writ of mandamus) or suspend a law (by injunction). This is because the executive branch IS the branch that enforces laws. The President can't complain about himself.
Who can determine that a law is unconstitutional?
Any court - state or federal - and only a court. A court's judgment is only limited to its jurisdiction, though. If the US 2nd Circuit declares that a law is unconstitutional, that law is unconstitutional in the New York area. In Georgia, it's still the law. Only the Supreme Court can declare a law unconstitutional finally and for the whole of the United States. Any lower court ruling can be appealed to a higher court with a larger geographic jurisdiction. There are many times where a law is declared unconstitutional in some state or circuit court and the federal government does NOT appeal for fear that a higher court will agree. The feds just quietly stop enforcing the law or only stop enforcing it in that area.
Does the President's oath mean anything?
No. Technically, the President doesn't even have to take an oath. He becomes President by operation of law. George W. Bush became President at 12:00 p.m., January 20, 2001, even though he didn't take the oath until several minutes later. If the Russians had attacked at 11:58, Clinton would have been in charge for two minutes, then Bush. And everybody, even the Russians, would have been surprised.
The President's oath to "support and defend the constitution" is meaningless. It is not a law. He cannot be held to the words of his oath nor punished for ignoring them. He might as well have receited the words to Blinded By The Light, assuming he could understand what Manfred Mann was saying.
Can a case go directly to the Supreme Court?
Almost never. Almost all cases start out in a trial court. If no facts are at issue, the sides might stipulate to a set of facts and get a decision more quickly but almost nothing starts at the Supreme Court. In fact, the Court almost doesn't have to hear any case it doesn't want to. Of the over 10,000 cases appealed to the Supreme Court every year, only about 180 are heard and decided.
Can the Supreme Court issue advisory opinions?
No.
Please let me know if there are any other questions.