I am asked this question many times - for good reason. Why DO students traditionally take Algebra I, then Geometry, and then Algebra II? Although it is certainly possible to have a successful year in Algebra II before taking Geometry, here are some thoughts to consider as you decide what is right for your student.
First, it is important to know that the material covered in Geometry does build on the concepts taught in Algebra I. It is not a total break from Algebra - it is an extension. For example, Algebra I students are taught to manipulate linear equations and are introduced to graphing. Geometry students go much deeper into the graphing of linear equations. Then, in Algebra II, students combine both skills and go even further into the topic. Geometry really does fit nicely between the Algebra courses.
However, if your student's Algebra skills are not very strong, they are going to struggle in Geometry. Students who have not fully mastered Algebra I skills should register for Algebra I so that they have an appropriate foundation to build on in order to be prepared for the upper level coursework.
Occasionally we have students who completed Algebra I but, for whatever reason, do not feel like they mastered the concepts. These students could choose to take Algebra II as their next course in order to continue working on the Algebra skills. The Algebra II class would be a fairly significant amount of work because some of the material that is presented as a review would be new material for them. However, with diligence and effort, it could be done.
One more thing to consider is the end goal of your student's high school math courses. If your student plans to pursue a STEM degree (science/technology/engineering/math), it will be important for them to have a solid background in mathematics, including Pre-Calculus with Trigonometry course. A student who takes Algebra II, then Geometry, and then Pre-Calculus with Trigonometry will find that there is a significant jump between Geometry and Pre-Calculus with Trig.
It is strongly recommended that students who plan to pursue a STEM degree follow the traditional path of Algebra I, Geometry, Algebra II, Pre-Calculus with Trig and, if there is time, either Calculus I or AP Calculus AB. However, a student who is not going to pursue a STEM degree can certainly choose Algebra I, Algebra II, Geometry, and Pre-Calculus for non-STEM majors as an appropriate order of coursework.
What math class should your student take after Algebra II? Most universities recommend that students take at least three and preferably four years of math. After Algebra II, students have completed three years, so that fourth year could be considered optional. But I would almost always recommend that a student continue with at least one more math class.
If your student is going to pursue a course in business or another degree that requires Calculus I for non-majors (aka Business Calculus) then the HSML Pre-Calculus for non-STEM majors, taught by Joy Combs, is the class for them.
If your student is going to pursue a STEM career (science/technology/engineering/math) then the HSML Pre-Calculus with Trigonometry course, taught by Brenda Barnett, is the best option.
Both HSML Pre-Calculus courses use university level textbooks and are an excellent way to encounter the material while still in a high school style setting. After completion of either course, a student could choose to re-take the same class in a university setting so as to "get their feet wet" in that setting without encountering new and difficult coursework right away or they could choose to go right into the Calculus course of their choosing, confident in the fact that they have been adequately prepared for that next course.