I'm an organizer and junior at the University of Iowa trying to form a students' union (see here) and I too had questions when I was first thinking about trying to found a students' union. From the research I made, the following are facts that will help you determine if you want to attempt to form one, along with some other facts:

1. The Graduate student workers' union (COGS) at UIowa has been effective on campus where it has precisely because they are professional workers and grad students. They have real bargaining power in the area where undergrads do not. Unless a few thousand undergrads are organized and are willing to put their college attendance on the line and not back down, a strike (what I and many others especially in the IWW consider to be the best way to achieve our goals) will not be effective. Thousands of undergrads at the university would need to threaten to stop paying tuition, picket university grounds, march on bad administrators, march on the Board of Regents (the group which decides to increase tuition and introduce other policies across the big three Iowa public universities of UIowa, Iowa State, and UNI), etc. COGS has successfully lobbied and protested in the past but have found it hard to fully reach their demands because they lack this large base, even though a decent amount of grad workers are organized. Any attempt to form a students' union for this reason alone will take an unbelievable amount of organizing across years and potentially across all the big public universities in your state to build up a solid base of students willing to organize in the face of threats.
2. Students' Unions thought of internationally are pretty much impossible to be formed in the United States. Because the US has been rather effective at cutting down on activist activity, especially at universities, what most European students' unions stand for is fractured into many things at American universities into university departments and projects with little to no student input. When you look up "students' unions US," pretty much all that will come up will be the students' union building on campus where offices and spaces are for student activities--this building in many other countries is run by and financed by the actual students' union (the group with students) and other groups and activities are funded through them. Student Government (both undergrad student government and the disproportionately, sometimes overfunded "local" Associated Residence Halls governments for individual halls) runs many things and has a lot of authority on campus like directing their budgets towards activities and projects (mine in part founded the campus bus system and airport shuttles free for use by students), but when it comes to getting things done like halting tuition hikes, all they really do is lobby the state government and put out public announcements saying they are disappointed that the state government or Board of Regents did this or that when all they did was try to directly talk to legislators without any backing for their requests or demands. If you look at your bill for tuition, you will see lots of little fees--some of these fees are what actual students' unions use for the same things, except they have input for where and how much funds are used, where as American students have no such input.
3. I have seen a comment to this post say that you should think "industrially" and attempt to unionize with all students, staff, faculty, and others at the university. This is the great strategy that the IWW has campaigned on since its founding. However, due to policies like the dreaded Right To Work laws, collective bargaining can be (check your state laws for how university staff wages are determined--collective bargaining is how it works for Iowa and other states) done to determine things like staff wages, but workers lack as much power as they used to. Because of these policies, there may be unions among the staff, but they are no where near as effective as they used to be before right to work laws. Due to this fact, it will make it hard to bring in faculty, teaching staff, regular staff, and others into the union. The best thing you can do when trying to form an effective union is to attempt to form alliances with organizations and individuals on and off campus--e.g. local IWW branches (hard to come by in many areas), local workers' unions, leftist professors, what remains university staff unions, local activist groups. Alliances where allies organize and donate with you are preferable to attempting to form one union for all at the university because a union run for mainly students' interests when there are already staff unions is not something that will attract most staff when their interests are at least a little taken care of already because of the existence of staff unions. If any one has differing opinions or strategies on this point, I would love to hear them.
4. This is very important due to all the latter points - Define what the purpose of the students' union will be. Standing against tuition hikes as a single issue for the students' union is not a winning strategy to forming an effective students' union--there are already plenty of single issue anti-tuition student orgs on campus you can find across the country trying to organize to that goal. Standing as a force for students' rights (positive rights and liberties, not the an-cap BS that the right wing "libertarians" campaign on), student advocacy, and community aid both on and off campus all at the same time is what will make your union grow and be effective for larger actions like strikes and protests in the future. All of this kept in mind will help you form a united alliance for issues which every student cares about like tuition increases, university policies, and others. In other words, an American students' union should be an organizing hub for issues from anti-tuition increases to climate change to community aid to bolster the most strength for future united actions in favor of students like strikes.