Which calculus class comes after pre-calculus?

Probably something like Calculus 1. If you want, you might be able to do AP Calc AB, but that might prove a bit difficult.

The typical timeline for maths is Precalculus, then Calculus I/AP Calculus AB, then Calculus II/AP Calculus BC (content that is exclusive to AP Calculus BC is what is taught in Calculus II. Everything else is from Caluclus I), then you would take Calculus III (sometimes referred to as Multivariable Calculus), Differential Equations, and Linear Algebra simultaneously.

Differential calculus, then Integral calculus.  Khan Academy has like 5 different Calc 1 courses, so it's confusing.

Calculus AB/BC will not prepare you for…anything.  Well, except Calc 1 in college, maybe—everybody in my Calc 1 class took Calc AB or BC in high school, and in a class of 300 the average grade 2/3 of the way into the course was 68%.  C (70%-79.99%) is failing, and that course is focused on knowing the content (about 20% of quizzes are dropped, 90% correct on homework is automatically awarded 100% credit, 10% of homework is dropped, and all exams including the final are graded out of 100 points but award up to 110 points).

The book for that class was terrible.  I just used Differential Calculus on Khan Academy, and the beginning of Integral Calculus.  A lot of people used other online resources.

My final overall grade was 95%.  I passed the placement test for that course by spending 6 weeks going through Khan Academy's College Algebra course; I want to say 80% of that material was new to me, but I recognized almost nothing from the College Algebra course, and I passed the AP Calc AB test.  High school algebra was garbage (this is what will kill you in Calc—you don't suck at math, you just never got taught right and don't have the foundational algebra needed to not totally fail Calc 1, go back and run through the algebra course again if you're struggling with Calc).

If you can handle differential and integral calculus no sweat, WELCOME TO ENGINEERING!  We've got more jobs opening every year than we have graduating engineers, I hope you like entry-level salaries that look like the mid range for an IT security professional with 15 years of experience because that's exactly what you're going to get the day you graduate college.

Thank you. I will start with Differential Calculus.