The individual agencies, under the guidance of a Cabinet secretary (appointed by the President) submit their budgets. It's a combination of the work of the career civil servants (largely continuing their existing work, plus some new things they'd like to do) and the goals of the President.

That's the budget sent to Congress, and traditionally pronounced "dead on arrival". In practice, the Congress is usually pretty interested in continuing to fund existing programs (many of them mandated by law), and changes (adding and removing stuff) is done around the edges.

The final appropriations bill (authorization to spend money) is passed by the House, then the Senate, and then possibly reconciled between the two.

That's then sent to the President, who can sign it or veto it. The veto could be overridden, but it's a very high bar to clear, and the President's party nearly always has enough votes to avoid that even if they're not an outright majority. So then there's a protracted negotiation... during which the government may shut down if the existing appropriations bill expires (including temporary extensions).