Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or have tangency of order $k$ (for any $k$ between 3 and $d$) with $C$? Probably this can be done for small $d$ using the equation for $C$, but I would like to find out if there is a formula for general $d$.

You have some polynomial $f(x,y,z)$. A line through the point $(1:0:0)$ can be paramaterized by a map from $\mathbb P^1: (u:v) \to (u:av:bv)$ for some constant $a$ and $b$. $f$ restricts to a degree $d$ polynomial in $u$ and $v$. Since it has no roots where $v=0$, set $v=1$. You now have a univariate polynomial such that the $k$th coefficient is a degree $k$ polynomial in $a$ and $b$. The discriminant of this polynomial determines whether it has a double root, which is the same as if the line is tangent. The discriminant is a degree $d(d1)$ polynomial in $a$ and $b$. Thus, the generic number of roots is $d(d1)$. This is similarly the generic number of tangent lines. The generic number of inflection points or higher is zero, as this would require a multiple root of the discriminant. To check that there is not some extra constraint on the discriminant polynomial that forces a multiple root, we can just find an example that does not have a multiple root. If $f(x,y,z)=x^d+xy^{d1}+z^d$, then $f(u)=u^d+ua^{d1}+b^d$, whose discriminant, which is $a^{d^2d}+b^{d^2d}$ up to some constant factors, has no multiple roots. 


Let $\tilde\pi:\mathbb{PC}^2\setminus\{P\}\to\mathbb{PC}$ be the projection from $P$, which restricts to a surjective morphism $\pi:C\to\mathbb{PC}$ of degree $d$. You are asking for the number of points of ramification of this morphism with multiplicity, or more precisely the degree of the ramification divisor $R$. By Hurwitz' theorem, it can be computed as $$\deg(R) = 2g(C)2  d(2g(\mathbb{PC})  2)=2\binom{d1}{2} + 2(d1) = d(d1).$$ 

