At least partial answers to both your questions may be found in the following paper:

Patricia Hersh and Isabella Novik, A short simplicial $h$-vector and the upper bound theorem, Discrete & Computational Geometry, 28 (3): 283-289 (2002).

In particular, the main theorem implies (via Corollary 1) that if the dimension $d$ is odd, then the first $d-1$ entries of your $f$-vector will be bounded above by those of the cyclic polytope $C(n,d)$. For additional details, look for the "upper bound conjecture" as well as the references in this paper.

I don't think much is known when $d$ is even.

As Gil remarks in his comment, Corollary 1 of the paper which I mentioned does not in fact imply the upper bound conjecture except when one additionally assumes isolated singularities. Still, I hope that this paper and its references may be of some use to the OP.

At least partial answers to both your questions may be found in the following paper:

Patricia Hersh and Isabella Novik, A short simplicial $h$-vector and the upper bound theorem, Discrete & Computational Geometry, 28 (3): 283-289 (2002).

In particular, the main theorem implies (via Corollary 1) that if the dimension $d$ is odd, then the first $d-1$ entries of your $f$-vector will be bounded above by those of the cyclic polytope $C(n,d)$. For additional details, look for the "upper bound conjecture" as well as the references in this paper.

I don't think much is known when $d$ is even.