What does the small, steep curve at the very beginning of a weak acid strong base titration curve come from?
This is part of estsblishing the buffer associated with the weak acid. To establish a buffer you need relatively large amounts of acid and conjugate base, and a weak acid can't produce the necessary conjugate base by itself.  You need to push it along with a little strong base to get into the buffering condition.

Say your solution is 0.1 molar acetic acid, with $K_\mathrm{a}=1.8×10^{-5}$. Before titration the pH is given by the Henderson-Hasselbach equation.  The logarithm of the above $K_\mathrm{a}$ value is about -4.74, therefore:

$\mathrm{pH}=4.74-\log_{10}\left(\frac{[\ce{HC2H3O2}]}{[\ce{C2H3O2-}]}\right)$

Now the amount of $\ce{C2H3O2-}$ in the solution is just what you get from dissociation of the acid, but acetic acid is only a weak acid and the actual amount of dissociation is small, in this case only about 1.3% of the acid you dissolved.  Merely 0.01 molar strong base reacting with the acid is enough to make a lot more acetate ion than just the acid dissociation alone, so the ratio in the H-H equation makes a big change and the $\mathrm{pH}$ follows suit.  Only when you have large amounts of both acid and conjugate base does the $\mathrm{pH}$ become buffered against slight additions of base or acid.