= Solution
A <quasi-coherent sheaf> is a <sheaf of modules> which locally has a presentation by <free module> <sheaves>:
$$
\mathcal O_U^{(J)}\longrightarrow\mathcal O_U^{(I)}
\longrightarrow\mathcal F|_U\longrightarrow0,
$$
where the index sets may be infinite. Equivalently, on affine neighbourhoods it is an <affine module sheaf>. A <coherent sheaf> is locally finitely generated, with the additional requirement that the kernel of every map from a <finite free module> <sheaf> is locally finitely generated. On a <Noetherian scheme>, these conditions are equivalent to being locally associated with a <finitely generated module>.
For an <affine variety> $X$ with <coordinate ring> $A$ and an $A$-<module> $M$, set
$$
\widetilde M(D(a))=M_a,\qquad
\widetilde M_P=M_{\mathfrak m_P}.
$$
Restriction on <principal open subsets> is the natural <module localization> map. On an arbitrary open subset, sections are families of these stalk values locally represented by a single fraction from a <localization>. This constructs the <affine module sheaf>; its <global sections> are $M$. Choose a free-module presentation $A^{(J)}\to A^{(I)}\to M\to0$. <Exactness of localization> and its compatibility with direct sums yield
$$
\mathcal O_X^{(J)}\longrightarrow\mathcal O_X^{(I)}
\longrightarrow\widetilde M\longrightarrow0.
$$
Thus $\widetilde M$ is a <quasi-coherent sheaf>, without a finite-generation assumption.
We now prove <coherence of an affine module sheaf>. If $M$ is finitely generated, take a <surjection> $A^r\to M$. Since $A$ is a <Noetherian ring>, its kernel is finitely generated, so $M$ has a <finite presentation of a module>. More generally, on every principal affine open, the kernel of a map $A_a^s\to M_a$ is a <submodule> of a <finite free module> over the <Noetherian ring> $A_a$, and is finitely generated. On sheafifying, this verifies both parts of the definition of a <coherent sheaf>.
Conversely, if $\widetilde M$ is coherent, choose a finite principal-open cover $X=\bigcup_iD(a_i)$ on which it has finitely many generators. Write those local generators as $m_{ij}/a_i^{r_{ij}}$. Their numerators generate a finite <submodule> $N=\sum_{i,j}Am_{ij}$ of $M$, and $N_{a_i}=M_{a_i}$ for every $i$. To check that $M/N=0$, let $q\in M/N$. Each equality after <localization> gives an exponent $r_i$ such that $a_i^{r_i}q=0$. The ideal generated by the $a_i^{r_i}$ is the <unit ideal>: otherwise a maximal ideal containing all of them would give a point outside the principal-open cover. A linear combination of these powers equal to $1$ consequently annihilates $q$. Every $q$ is zero, proving
$$
\boxed{\widetilde M\text{ is coherent}\quad\Longleftrightarrow\quad
M\text{ is a finitely generated }A\text{-module}.}
$$
Next consider a morphism $\phi:\mathcal G_1\to\mathcal G_2$ between <quasi-coherent sheaves>. On an affine open $V$ of the target <variety>, use the assumed <affine module-sheaf equivalence> to write $\mathcal G_j|_V=\widetilde{M_j}$, with $M_j=\Gamma(V,\mathcal G_j)$. The <sheaf> morphism is the sheafification of the <module> map $\psi=\Gamma(V,\phi)$. Indeed, its action on a section $m/a^r$ of a principal open is forced by module-linearity and restriction to be $\psi(m)/a^r$. Therefore <exactness of localization> gives
$$
(\ker\phi)(D(a))=\ker\bigl((M_1)_a\to(M_2)_a\bigr)
=(\ker\psi)_a.
$$
These identities are compatible with restriction, so $(\ker\phi)|_V=\widetilde{\ker\psi}$. Since this holds on an affine cover, \b[the <kernel sheaf> is quasi-coherent on every <variety>]. This proof of <kernels of quasi-coherent sheaf morphisms> does not assert that taking <global sections> is right exact.
For <direct image> along a morphism of <affine varieties>, write $Y$ with <coordinate ring> $A$, $X$ with <coordinate ring> $B$, and let $\alpha:A\to B$ be the induced ring homomorphism. If $\mathcal F=\widetilde M$ on $X$, regard $M$ as an $A$-module by <restriction of scalars>. For $a\in A$,
$$
f^{-1}(D(a))=D(\alpha(a)),\qquad
(f_*\mathcal F)(D(a))=M_{\alpha(a)}=M_a.
$$
The last equality identifies <localization> with the scalar action of $a$. It identifies the <direct image sheaf> with the <affine module sheaf> of the restricted <module>, so $f_*\mathcal F$ is quasi-coherent.
Suppose $Y$ is affine and $X=\bigcup_{i=1}^rU_i$ is a finite affine cover. Since a <variety> is separated, every $U_i\cap U_j$ is affine: it is identified with the inverse image of the closed <diagonal morphism> inside the affine product $U_i\times_kU_j$. Write $g_i=f|_{U_i}$ and $g_{ij}=f|_{U_i\cap U_j}$. Form the morphism of <sheaves of modules>
$$
d:\bigoplus_i(g_i)_*(\mathcal F|_{U_i})
\longrightarrow
\bigoplus_{i,j}(g_{ij})_*(\mathcal F|_{U_i\cap U_j}),
\qquad
(s_i)\longmapsto(s_i|_{U_i\cap U_j}-s_j|_{U_i\cap U_j}).
$$
Both sums are finite, and every summand is quasi-coherent by the affine calculation. For every open $V\subseteq Y$, the <sheaf gluing axiom> on $f^{-1}(V)$ identifies the kernel of this map on sections with $\mathcal F(f^{-1}(V))$. Hence
$$
f_*\mathcal F=\ker d.
$$
The kernel result proved above establishes quasi-coherence.
Finally, for arbitrary $Y$, take an affine neighbourhood $V\subseteq Y$ and apply the preceding argument to $f^{-1}(V)\to V$. The open subset $f^{-1}(V)$ is again a <variety> and has a finite affine cover, because <varieties> have <Noetherian Zariski topology>. The restriction of $f_*\mathcal F$ to $V$ is exactly the direct image for this restricted morphism. Thus
$$
\boxed{f_*\mathcal F\text{ is quasi-coherent for every morphism of varieties}.}
$$
The finiteness and separation conditions are the relevant hypotheses, not affineness of the original source or target. More generally, <quasi-coherence of direct image under a quasi-compact quasi-separated morphism> is valid. If one permits nonseparated finite-type prevarieties, each overlap in the argument has a finite affine cover; replacing its summand by the sum of direct images from these charts gives the same kernel, since agreement can be tested on a cover. This also proves the assertion in that convention. It does not extend by this argument to arbitrary spaces with an unrestricted infinite cover.
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