If $s$ is a smooth section of a vector bundle $V$ and $t$ is a smooth section of a vector bundle $W$ then $s \otimes t$ is a smooth section of $V \otimes W$. Its value at each point $m \in M$ in the underlying manifold $M$ is $(s \otimes t)(m)=s(m) \otimes t(m)$. I am still not sure if that is what you are looking for. This has nothing to do with connections, so I don't see why you mention connections in your question. If $s_1, s_2, \dots, s_p$ is a basis of local smooth sections of $V$ over an open set $U$ (i.e. these sections are defined on $U \subset M$ and every local smooth section defined on $U$ is a unique linear combination $\sum_i f_i s_i$ with $f_i$ smooth functions) and similarly $t_1, t_2, \dots, t_q$ is a basis of local sections of $W$ over the same open set $U$, then $s_i \otimes t_j$ for $i=1,2,\dots,p$ and $j=1,2,\dots,q$ is a basis of local smooth sections of $V \otimes W$ over $U$. So that should explain what all of the sections look like, I hope.