Let $\{V_i: i \in I\}$ be an infinite collection of non-trivial (i.e. not dimension $0$) vector spaces. I am trying to prove that $\bigoplus_{i \in I} V_i$ is not isomorphic to $\Pi_{i \in I} V_i$.
Here is what I tried: Towards a contradiction, assume there exists an isomorphism $f:\bigoplus_{i \in I} V_i \to \Pi_{i \in I} V_i$. Then choose a basis $B=\{b_j\}_j$ for $\bigoplus_{i \in I} V_i$. Observe that $C=\{f(b_j)\}_j$ must then be a basis for $\Pi_{i \in I} V_i$.
To reach the desired contradiction, I tried to come up with a diagonalization-style argument to choose a vector $w \in \Pi_{i \in I} V_i$ such that $w$ cannot possibly be a finite linear combination of $f(b_j)$ terms. But it's harder than I thought. For example, if we are lucky enough that each $f(b_j)$ is $0$ in all but finitely many coordinates, then we can just choose $w$ to have all of its coordinates nonzero and we're done. But in general $f(b_j)$ might not be so nice so it's harder.
(Please let me know if anything in my question needs clarification)
In fact, the result you're trying to prove is not true in general! For example, taking $I=\mathbb{N}$ and setting all the $V_i$s to be the same for simplicity, we can have a vector space $V$ such that $$\prod_{i\in\mathbb{N}}V\cong \bigoplus_{i\in\mathbb{N}} V\cong V,$$ essentially for the same reason that we can have an infinite cardinal $\kappa$ such that $$\kappa^{\aleph_0}=\kappa.$$ This happens if e.g. $\dim(V)=2^{\aleph_0}$, so as a concrete example if our field of scalars is $\mathbb{R}$ we can take the vector space of all (not necessarily continuous!) functions $\mathbb{R}\rightarrow\mathbb{R}$.
(If the $V_i$s are all finite-dimensional vector spaces, then the result will hold since we will have $\dim(\bigoplus_{i\in I}V_i)=\vert I\vert$ but $\dim(\prod_{i\in I}V_i)=2^{\vert I\vert}.$)
Note that thinking in terms of dimension calculations simplifies things substantially; since a vector space is determined up to isomorphism (once we fix the field of scalars) by its dimension, which is a cardinal, for questions like this we're more-or-less just doing straight set-theoretic combinatorics in disguise.