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.