As coordinator of a Maths Learning Centre, where many many students have actually seen some calculus at school but still struggle with calculus at University, these are my top few:
The notation f(x)
Quite a few students think that $f(x)$ means $f \times (x)$. This is particularly true with trigonometric functions, where many of them actually say "sine times ex" when they read $\sin(x)$ aloud. I teach them it's pronounced "sine of ex", but for some that doesn't help because they were taught at school that "of" always means "times" (as in "10% of 15"). Constantly reinforcing that $f(x)$ is the object produced when the action $f$ is performed upon the object $x$ -- or "the result of f acted upon x" -- would make life better for many of them.
Even if students do realise that $f(x)$ is not about multiplication, many of them do not know at a deep level that $f(x)$ is actually a number. They see it as a call to action, rather than the name of the result of this call to action. While it may be possible to calculate the value of $f(3)$ and it might come out to the answer $4$, say, you don't necessarily need to calculate it, and you can use the symbol "$f(3)$" every time you mean "$4$" if you want. Basically, $f(3)$ is literally able to stand in the place of whatever value it is.
This may seem like a small issue, but it is a problem when students attempt to differentiate $\ln(2)$ to give $\frac12$ (for example, given $y = e^{\ln(2)x}$, they write $\frac{dy}{dx} = e^{\ln(2)x}(\frac12 x + \ln(2))$ ). I keep reinforcing to them that $\ln(2)$ is just a number. It's somewhere between 0 and 1, but I don't have to calculate it to know it's a number.
The fundamental meaning of an x-y equation
I really wish students knew that the fundamental meaning of an x-y equation is to tell if a point is part of set in 2D space. That is, when we see something like $y = 3x+1$, it is telling us a rule for how x and y must be related in order for the point $(x,y)$ to be on a line. If we sub in the point and it works, then it's on the line, and if it doesn't work, then it's not on the line. The question of "Is the point (1,4) on the line with equation $y = 3x+1$?" stumps many of my students, even if they can tell me what the slope and intercepts are without even thinking.
Another point I'd like them to realise is that any equation such that all the points on the line satisfy it and all the points not on the line don't satisfy it, is a possible equation of the line. For example, $y - 3x = 1$ and $x = \frac{y-1}{3}$ or even $(y - 3x +1)^2 = 0$ are perfectly reasonable equations of the same line.
Basically my point is that I would like students to know that an equation does not have to be explicitly tied to a function, and even if it is, it can still be used to tell if points are part of a set.
The connection between algebra and geometry
The flip-side of the above point is that equations do represent geometrical objects, and you can glean information from the equation to tell you about the geometry of the object. I guess what I'd like them to realise is that writing an equation in a certain way will tell you certain information about the object, and especially that different ways of writing the equation will tell you different information. These ideas are often known separately for different equations (eg quadratic equations where completeing the square tells you the vertex but factorising tells you the x-intercepts), but I'd love them to have the general concept reinforced for everything so that they realise it's actually one idea worked out in different situations.
Knowing what the graphs of various functions look like
My students self-identify this one as their own major problem. They just don't have an instinct for what the graph of a function looks like without having to get a computer/graphics calculator draw it for them. If you ask them "what does the graph of a cubic look like?", they will reach for their graphics calculator, when all you wanted them to do was draw a "swoosh".
I would love it if they simply "knew" what various graphs looked like. A bit of a stimulus-response thing where if they see either of the forms of the equation of a line, they'll think "line", or if they see $y = \text{quadratic in }x$ they'll automatically think of an up-curve or a down-curve.
I realise this may be at odds with the previous point, but I think you can achieve both. Plus it really is an issue for them when their university lecturers use these functions as their basic examples, and the students simply can't imagine them!
Skill at algebraic manipulation
Finally, this really is a prerequisite that gets in the way for a lot of people. It's not as bad a problem as you might think, because often learning algebra "in passing" while doing calculus is actually more successful than the students' previous algebra instruction. However, it would be nice if they knew how to expand/factorise/manipulate fractions.
This last one of manipulating fractions particularly comes up when they are doing differentiation by first principles with limits, and when they are using the quotient rule, and many students show that they simply do not know how to add fractions, even of the number variety. (Though there are some students who can do the algebraic ones and not the number ones.)
big-list
tag? On the one hand, this question would be perfect for it; on the other, I don't know that we should in general encourage lots ofbig-list
type questions. $\endgroup$