Your understanding is incorrect in both cases.

A Taylor series is able to represent an "arbitrary" function $f$ in the neigbourhood of a given point $a$ in the domain of $f$ as a power series: $$f(x)=\sum_{k=0}^\infty c_k (x-a)^k\ ,$$ i.e., in the form of an "infinite polynomial". Neither the series nor its finite partial sums are "linear functions". The coefficients $c_k$ of this power series are connected to the represented function $f$ by the formula $c_k=f^{(k)}(a)/k!\ $. So we see here the values $f^{(k)}(a)$ entering in a linear way, but the derivatives $f^{(k)}$ as functions do not appear in the representation.

A Fourier series is able to represent an "arbitrary" periodic function $f$ of period $2\pi$ as an "infinite linear combination" of the basic periodic functions $t\mapsto \sin(k t)$, $t\mapsto \cos(k t)$; so it has the form
$$f(t)={a_0\over 2}+\sum_{k=1}^\infty (a_k\cos(kt)+b_k\sin(kt))\ .$$
The coefficients $a_k$, $b_k$ in this representation are connected to the given $f$ via certain integrals (which I won't write down here).

In fact there is a certain connection between these two paradigms. It works in the realm of functions of a complex variable $z$, but there is no question of the same function $f$ of a real variable $x$ or $t$ being represented with more or less the same coefficients $c_k$, $a_k$, $b_k$ first as a Taylor series and then as a Fourier series.