Let me answer your last question "How can I think geometrically (in the lattice) about fixing a polarization?". I will follow the treatment given in [Birkenhake-Lange, *Complex Abelian Varieties*, Chapter 3].

Let $X = V / \Lambda$ be a complex torus of dimension $g$ and $L$ a line bundle on $X$ with first Chern class $H$. Then $H$ is an Hermitian form on $V$, whose alternating form $E:= \textrm{Im } H$ is integer-valued on the lattice $\Lambda$.

By standard linear algebra there is a basis $\lambda_i, \ldots, \lambda_g$, $\mu_1, \ldots, \mu_g$ of $\Lambda$, with respect to which $E$ is represented by the matrix
$$D=\begin{pmatrix} 0 & D \cr -D & 0 \end{pmatrix},$$
where $D=\textrm{diag}(d_1, \ldots, d_g)$ and the $d_i$ are non-negative integers satisfying $d_i | d_{i+1}$. Moreover, the $d_i$ are uniquely determined by $E$ and $\Lambda$ and thus by $L$.

The vector $(d_1, \ldots, d_g)$ is called the *type* of $L$; the line bundle $L$ is a polarization (i.e, $L$ is ample) if and only if all the $d_i$ are strictly positive. The basis $\lambda_i, \ldots, \lambda_g$, $\mu_1, \ldots, \mu_g$ is called a *symplectic basis* for $\Lambda$. Setting $$\Lambda_1 := \langle \lambda_1, \ldots, \lambda_g \rangle, \quad \Lambda_2 := \langle \mu_1, \ldots, \mu_g \rangle$$
we obtain a decomposition $$\Lambda = \Lambda_1 \oplus \Lambda_2,$$
where the $\Lambda_i$ are isotropic with respect to $E$.

Finally, the Riemann conditions can be expressed as follows: set $e_j= \lambda_j /d_j$ for $j = 1, \ldots, g.$ Then $\mathscr{B} = \{e_1, \ldots , e_g \}$ is a basis for $V$, and with respect to this basis the lattice $\Lambda$ can be written as $$\Lambda = \tau \mathbf{Z}^g \oplus D \mathbf{Z}^g,$$
where $\tau$ is a complex, symmetric square matrix of order $g$ whose imaginary part is positive defined. From this, it follows that the moduli space of abelian varieties with polarization of type $(d_1, \ldots, d_g)$ is a quotient $$\mathcal{A}_{g, D} = G_D \backslash \mathscr{H}_g,$$
where $$\mathscr{H}_g :=\{ \tau \in M_{g \times g}(\mathbf{C}) \, | \, \tau = {}^t\tau, \, \, \textrm{Im }\tau >0 \}$$
is the Siegel upper half-space and $G_{D}$ is a suitable discrete subgroup of the symplectic group $\textrm{Sp}_{2g}(\mathbf{R})$. Here the left action of $\textrm{Sp}_{2g}(\mathbf{R})$ on $\mathscr{H}_g$ is the natural one, namely $$\begin{pmatrix} a & b \cr c & d \end{pmatrix} \cdot \tau := (a \tau + b)(c \tau + d)^{-1}.$$