What's the best way for a student in my situation to get this fixed so the school stops wasting student's time with outdated and wrong information?
I am not a student, but an adjunct faculty member. That said, as an adjunct I have very little official authority when it comes to curriculum matters, but I have managed to find success in updating woefully outdated curriculum. I will share some things based on my experience.
First, I recommend you start by talking with the instructor(s) of the class(es) in question. Request a copy of the master syllabus. This is not the master copy of the syllabus handed out each term, but is actually a specially formatted and fairly detailed documented that describes precisely how the course will meet all of the educational objectives required by the school (and/or any accrediting body). Also get a copy of the course catalog description (you can probably get this yourself from your school catalog online). Also ask the instructor if they are willing to support your effort and advocate for the change before any school personnel or committees involved in the process (as a student you may not be permitted to appear before those personnel or committees to request a curriculum change). Find out if there are minor changes that you can make that satisfy your objective for updating the course without triggering a full academic revision of the course. There may processes in place for smaller changes that require lesser review and approval.
Then, get to work. Update the master syllabus to reflect what you think would be a better course composition, sequence of topics, etc. Ensure that all required school objectives are still being met or exceeded. Provide supporting documentation. That might include attachments that describe academic developments in the field, analyses about emerging new topics that are shaping the field, etc. Throughout the process work closely with the instructor involved. If you are fortunate enough to be able to interact directly with the course director, then you will have fewer layers to go through. If not, the instructor you are working with will need to make a proposal to the course director, probably the department, and either an undergraduate or graduate committee that reviews and approves curriculum changes.
You will need to ensure to get buy in from the instructor involved and/or the course director as appropriate before the matter will come before the right committee. Offer to be a TA for the updated course to help get things off to a good start. Offer to write up lecture notes and slides for the new material, offer to write sample homeworks, projects, quizzes, exams, etc., as appropriate for the subject matter.
You will also need to patient. Keep in mind that for traditional semester schools, Spring registration is already underway (meaning your change would almost certainly not be considered for Spring) and Fall registration will probably open sometime in February or March. That means that if you want to get a course updated for the Fall of next year (which would be the earliest possible update if you started working on it today), you probably only have something like a month to get it all in order. Between Thanksgiving, Christmas/New Year's break, and Spring semester start up, you really don't have a big time window to get the job done, so you would have to hustle to have a chance.
That said, be prepared to wait as well. The committees probably only meet every few months, so you may have to request a special review for something to make it in time for next Fall. That may or may not be feasible depending on your institution and its policies.
I hear plenty of students complain about stuff like this, but I have not yet seen one actually try to tackle the problem rather than just complain about it. Best of luck.