It would be filed in the US District Court for the District of Columbia - the DC federal court. It wold be assigned to one of the judges of that court.
Those joining as plaintiffs would likely hire a DC law firm to represent them. The president would be represented by the White House Office of General Counsel which may, or may not, hire outside counsel.
After the Complaint is filed, there will be a preliminary hearing to set a schedule. After that, there will be a number of motions to dismiss filed by the White House. The plaintiffs will respond to these motions, and a hearing date will be set for arguing them.
During this time, several other things will happen. A parade of loons will file motions to join the case for various reasons, or at least will file briefs in support of the plaintiffs and ask the court to consider the arguments of putatively interested non-parties. Also, the plaintiffs will serve a number of discovery requests seeking to go fishing in the White House. The White House will not respond to these requests, arguing they are pointless until the motions to dismiss are heard and ruled upon. The plaintiffs will argue they need some discovery in order to address the motions to dismiss. A magistrate judge will be assigned to decide the discovery dispute, or at least to prepare a recommendation for the judge.
After all these fun & games, the motions to dismiss will be heard. The case will be dismissed for any or all of the grab bag of reasons for which it should obviously be dismissed - standing, political question, justiciability, and so on.
As to the "what can they win" question, the answer is pretty much a nothing burger even if they do win. The basis of their suit is a delay in the employer provisions of the ACA. The delay will actually be over by the time this case will come to a full parade rest, so the "remedy" will be moot at that time.
As far as a time line goes, you first have to appreciate that a federal official being sued in his or her official capacity has (and this is off the top of my head from FRCP Rule 4) I think 60 days to file an answer or motion to dismiss, from the time the suit is served. In other words, if the case were filed and served tomorrow, the motions to dismiss wouldn't be due until some time in October. If I recall correctly, the employer provisions were delayed until October. So it could easily become moot, in terms of any conceivable remedy, before an answer or motion to dismiss is actually due.
Some other procedural wrinkles are that the plaintiff can amend the complaint at least once without leave of court, just to keep the other side hopping. If they shoot for the moon, the case could be filed with a request for some sort of preliminary order on an expedited basis. That would be somewhat more amusing, because those sorts of motions are decided on a "quick look" basis to assess whether there is a serious ongoing harm and that the plaintiff has a strong likelihood of success down the road. That would at least give the plaintiffs something to run up through the DC Circuit Court of Appeals on the way to.the Supreme Court. That's even trickier since rulings on those kinds of filings are granted a higher degree of discretion with the district court from the get go.
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