Denote by $T_0,T_1$ a basis for the global sections of $\mathcal{O}_{\mathbb{F}_1}(f)$.  Denote by $X$ a basis for the global sections of $\mathcal{O}_{\mathbb{F}_1}(C_0)$.  Denote by $Y$ a global section of $\mathcal{O}_{\mathbb{F}_1}(C_0+f)$ that is linearly independent from $T_0X,T_1X$.  Then the "total coordinate ring" or "Cox ring" of $\mathbb{F}_1$ is $S_*(\mathbb{F}_1) = k[T_0,T_1,X,Y]$.  This is a $\mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}$-graded ring where you can think of elements in $\mathbb{Z}\times \mathbb{Z}$ as linear combinations $af+bC_0$.  Thus the degree of $T_0,T_1$ is $1f+0C_0$, the degree of $X$ is $0f+1C_0$, and the degree of $Y$ is $1f+1C_0$.  The embedding of $\mathbb{F}_1 \hookrightarrow \mathbb{P}^{17}$ induces a graded algebra homomorphism $S_*(\mathbb{P}^{17}) \to S_*(\mathbb{F}_1)$, i.e., 
$$k[Z_{l_0,l_1,m,n}|(l_0,l_1,m,n) \in I] \to k[T_0,T_1,X,Y], \ Z_{l_0,l_1,m,n} \mapsto T_0^{l_0}T_1^{l_1}X^mY^n.$$
Here, by definition, $I$ is the set of $4$-tuples of nonegative integers such that 
$$(l_0+l_1)(1f+0C_0) + m(0f+1C_0) + n(1f+1C_0) = 5f+3C_0, $$ 
i.e., $l_0+l_1+n$ equals $5$ and $m+n$ equals $3$.  In particular, this is a monomial homomorphism, or equivalently, a map of semigroup rings coming from the map of semigroups $\phi:\mathbb{Z}_{\geq 0}^{18} \mapsto \mathbb{Z}_{\geq 0}^4$ as indicated.  Therefore the kernel is generated by the  rank 4, quadratic binomials,
$$
Z_{l_0,l_1,m,n}Z_{l_0',l_1',m',n'} - Z_{l_0'',l_1'',m'',n''}Z_{l_0''',l_1''',m''',n'''},
$$
$$
\phi(l_0+l_0',l_1+l_1',m+m',n+n') = \phi(l_0''+l_0''', l_1''+l_1''', m''+m''',n''+n''').
$$

$\textbf{Edit}.$  I realize now that I am not certain whether these binomials are sufficient to generate the kernel.  Certainly the kernel is generated by binomials, and the kernel includes these binomials.  I believe these binomials define the closed subscheme, i.e., the saturation of the ideal generated by these binomials equals the full kernel.  However, I do not immediately see how to check that there are no further binomial generators.  Eisenbud and Sturmfels do give algorithms to check whether a binomial ideal is prime; this might do the trick.