This is my understanding of this yoga. It may not be exactly what you seek and may differ from another person's point of view. Also I apologize for my bad english.
For Grothendieck, many things should have a relative version. So instead of considering just a space $X_0$, consider a morphism $f:X\rightarrow S$ thought as a family of spaces $s\mapsto X_s:=f^{-1}(s)$.
Here is an easy example of relative thinking : Say you want to attach numerical invariants to your first space $X_0$, for example its dimension, you should instead replace it by a function on $S$, and you get the relative dimension. Under some assumptions, this function behave quite nicely.
If now you want to attach invariants that are sets, or abelian groups, to your first space $X_0$. Well there is a very nice tool that do exactly what you want for the relative version : presheaves on $S$. And under some assumptions, these are closely related to sheaves.
The six operations arise this way if you want to study cohomology.
There are many cohomology theories out there, and many of them have their 6 operations. But they do not behave exactly of the same way, so let's say we work with ordinary cohomology of (nice) topological spaces.
It is also better to work with derived categories. Indeed, the cohomology of space is in fact a complex up to quasi-isomorphism, more than just the groups $H^i$. Also, for example, the functor $f^!$ exists only at the level of derived categories. If $X$ is a space, let $D(X)$ be the derived category of sheaves on $X$ and let $\mathbb{Z}_X$ be the constant sheaf with value $\mathbb{Z}$ on $X$.
Let start with the first functor $f_*$ or rather $Rf_*:D(X)\rightarrow D(S)$. Well, this is exactly the functor that computes cohomology. Indeed, if $f$ is proper, $(R^if_*\mathbb{Z}_X)_s=H^i(X_s,\mathbb{Z})$ (this is the so-called proper base change theorem). So this is exactly the relative version of cohomology. In fact, if $f$ is proper and smooth (=submersion), then $R^if_*\mathbb{Z}$ are local systems on $S$.
There is also $f_!$ or rather $Rf_!:D(X)\rightarrow D(S)$. It does the same thing, but with cohomology with compact support : $(R^if_!\mathbb{Z}_X)_s=H^i_c(X_s,\mathbb{Z})$. This times, one does not need to assume $f$ proper.
And of course the tensor product $\otimes$ gives the multiplicative structure on cohomology. One can then speak of derived analogue of the Kunneth formula...
Now, if $g:T\rightarrow S$, there is the functor $g^*:D(S)\rightarrow D(T)$. In some sense this is the one that justify the whole thing : this is the functor which correspond to changing the base from $S$ to $T$. For example, if $j:U\rightarrow S$ is the inclusion of an open subset, one can form $X_U=f^{-1}(U)$ and $j^*Rf_*\mathbb{Z}_X=R(f_{|X_U})_*\mathbb{Z}_{X_U}$. This is a (very easy) special case of a very deep one : the smooth base-change theorem. Changing bases is VERY useful. For example, base change to the universal cover, so that local systems become constant. Or in algebraic geometry, base change to the algebraic closure. And of course, taking stalks are already special cases of base change...
And finally, the functor $f^!:D(S)\rightarrow D(X)$ and the internal Hom are there to deal with duality. (Note that $f^!$ is right adjoint to $f_!$, not left). Instead of a global duality between $H^i_c$ and $H_{d-i}$, we now have local versions, allowing local computations and so on... Just for completeness, if $S$ is a point, $f_*f^!\mathbb{Z}$ is the Borel-Moore homology and $f_!f^!\mathbb{Z}$ is the ordinary homology (at least if $X$ is nice enough).
By definition, a cosheaf on a space $X$ with values in a category $\newcommand\C{\mathcal{C}}\C$ is a sheaf with values in $\newcommand\op{\text{op}}\C^\op$. Thus to understand cosheaves, it suffices to understand sheaves.
In particular, to address your specific question, we have the following result.
Let $B$ be a base for the topology on $X$. Define the category of sheaves on $B$ in the usual way. Recall that sheaves on $B$ are functors $F$ from the opposite of the poset category of $B$ to $\C$ such that $$F(U)= \lim_{V\subseteq U} F(V),$$
where $U\in B$ and $V$ runs across basic open subsets of $U$. Note that I've written $=$ because $F(U)$ comes with a canonical cone given by the restriction maps.
Proposition. There is a fully-faithful functor
$$\C\newcommand\Shvs{\text{-}\mathbf{Shvs}}\Shvs(X)\to \C\Shvs(B),$$
induced by the restriction of a sheaf on $X$ to the open subsets in $B$.
Moreover, if $\C$ is complete, then this is an equivalence of categories.
Proof.
We need to show that the restriction functor is always fully-faithful and is essentially surjective if $\C$ is complete.
Let $F,G$ be sheaves on $X$. Let $F_B$, $G_B$ denote the restriction of $F$ and $G$ to the basis $B$. We know that if $U$ is any open subset in $X$, then since $F$ is a sheaf,
$$F(U) = \lim_{V\subseteq U, V\in B} F_B(V).$$
So if $\phi :F_B\to G_B$, then $\phi$ induces maps $F(U)\to G(U)$ for all open sets $U$ subset of $X$ compatible with the restriction maps. In other words, $\phi$ extends to a morphism $\phi' : F\to G$ which restricts to $\phi$ (not hard to check) on $B$. Moreover, this extension is unique, by the universal property of the limit.
This proves that the restriction functor is fully-faithful. (Full because all morphisms of $B$-sheaves can be extended, and faithful because the extension is unique).
Now if $\C$ is complete, if $F_B$ is a sheaf on $B$, then we can define
$$F(U) = \lim_{V\subseteq U, V\in B} F_B(V),$$
(we need completeness to guarantee that the limit exists), and you can check that this
defines a sheaf and $F(V)=F_B(V)$ when $V\in V$. Thus the restriction of $F$ to $B$ is (canonically isomorphic to) $F_B$. Therefore the restriction functor is essentially surjective if $\C$ is complete, and thus an equivalence of categories. $\blacksquare$
In particular, when we want cosheaves valued in abelian groups, these are the same as sheaves valued in $\mathbf{Ab}^\op$, and $\newcommand\Ab{\mathbf{Ab}}\Ab$ is cocomplete, so $\Ab^\op$ is complete. Thus this proposition applies. Cosheaves on a space are equivalent to cosheaves on a basis for that space.
Best Answer
A few things to note. First, a sheaf on a space $X$ is just a functor $P \in Target^{\mathcal{O}_X^{op}}$ such that for all open covers $\bigcup\limits_{i \in } U_i = U$, the obvious diagram $P(U) \to \prod\limits_{i \in I} P(U_i) \rightrightarrows \prod\limits_{j \in I} \prod\limits_{k \in I} P(U_j \cap U_k)$ is an equalizer.
This means that the definition makes sense in any category $Target$ which has small products. This is all we need for the definition of a sheaf.
Edit: in the comments, Eric Wofsey pointed out that $Target$ having products isn't even necessary. We can instead consider the diagram formed by the individual $P(U_i)$ and $P(U_j \cap U_k)$ and assert that $P(U)$, together with restriction maps, is the limit of this diagram.
Things get rather interesting when we consider the special case $Target = Sets$. In this case, we have an adjunction $a \dashv i : Sh(X) \rightleftarrows Sets^{\mathcal{O}_X^{op}}$, with $a$ being the sheafification functor. Moreover, $a$ preserves finite limits.
We can use the fact that $a$ preserves finite limits to show that for any equational algebraic theory with finitary operations $T$, a sheaf over models of $T$ is just a $T$-object in the category of sheaves.
To be more precise, a sheaf of groups is just a group object in the category of sheaves, a sheaf of left $R$-modules is just an $R$-module object in the category of sheaves, a sheaf of monoids is just a monoid object in the category of sheaves, etc.
This means that to prove things about sheaves of (groups, rings, modules, etc.), we just need to prove things about sheaves of sets with a (group, ring, module, etc.)-structure. This dramatically simplifies things in many cases.
It is especially nice if you use the full power of the category of sheaves as a Grothendieck Topos. This means that you automatically get most constructions that can be performed in set theory for free. Most basic constructions (free groups, coequalizers of modules, kernels and images, etc) are constructions which can be performed in any topos trivially, so they carry over to $Sh(X)$ with no effort whatsoever. Those constructions which are less basic (eg constructions of infinitary coproducts) apply in any complete/cocomplete topos.
Edit: for the sheafification construction, we just go from $Models(T)^{\mathcal{O}_X^{op}} \to T$-objects in $Sets^{\mathcal{O}_X^{op}} \to T$-objects in $Sh(X)$ (using ordinary sheafification, which preserves finite limits and hence $T$-objects) $\to T$-valued sheaves on $X$.