Q: Does *p++ increment p, or what it points to?
A: The postfix ++ and -- operators essentially have higher precedence than the prefix unary operators. Therefore, *p++ is equivalent to *(p++); it increments p, and returns the value which p pointed to before p was incremented. To increment the value pointed to by p, use (*p)++ (or perhaps ++*p, if the evaluation order of the side effect doesn't matter).
K&R1 Sec. 5.1 p. 91
K&R2 Sec. 5.1 p. 95
ISO Sec. 6.3.2, Sec. 6.3.3
H&S Sec. 7.4.4 pp. 192-3, Sec. 7.5 p. 193, Secs. 7.5.7,7.5.8 pp. 199-200