To expand the Supreme Court, they would have to pass a law. To pass a law under current rules requires
- A majority of those voting in the House of Representatives.
- A 60% super majority of those voting in the Senate to end debate.
- A majority of those voting in the Senate (which seem trivial to get if they previously had 60% to end debate).
- The signature of the president.
And if the president vetoes rather than signs, overriding the veto requires
- Two thirds of those voting in the House of Representatives.
- Two thirds of those voting in the Senate.
The 60% super majority of those voting in the Senate is set in the Senate rules. The Senate rules can be changed by a simple majority.
The two thirds super majority to override a veto is set in the constitution. To amend the constitution also requires two thirds of each chamber, or a constitutional convention called by the states. So it would be difficult to do this without the president's consent.
There is no requirement other than passing the bill though. If a Democratic president changed the system to allow a Democratic pack, the next Republican could pass a Republican pack. Because that's a bad idea, both sides have stuck with the current system. It's possible that one or more Senators who caucus with the Democrats might not support a court packing scheme that explicitly favors Democrats. In particular, Joe Manchin and Angus King come to mind. There may be others.
Note that reducing the court is harder. The last time it was done, they simply didn't replace retiring justices until the court was small enough. So a future Republican would be more likely to expand the court again than to start by shrinking it. They might expand it and then pass a law that would shrink it later.
Judicial term limits at the federal level would require a constitutional amendment.
There may be clever workarounds. For example, perhaps only nine justices would be allowed to vote on any particular issue. So there might be twenty-one justices (assuming Democrats and Republicans each add six), but only the nine longest serving unretired justices can vote at any one time. But that's going into untested waters and may be rejected as unconstitutional.