My question was a bit misguided, so I will try to clarify things as well as give some answers. A primary reference is Fulton, Intersection Theory on Toric Varieties.
Because any weighted projective space $P = \mathbb{P}(a_0, \ldots, a_n)$ which is non-trivial (i.e., one of the $a_i$ is greater than $1$) is singular, here we must first distinguish between the Chow homology groups $A_k(P)$, whose members are rational equivalence classes of $k$-dimensional cycles in $P$, and the Chow cohomology groups $A^k(P)$, whose members are homomorphisms acting on Chow homology (and decreasing degree by $k$). The Chow cohomology groups $A^k(P)$ assemble into a graded ring $A^*(P)$, whose product is given by composition; this ring acts on the module $A_*(P)$.
The Fulton paper describes both $A_k(P)$ and $A^k(P)$ in the case that $P$ is a general toric variety. Specializing to the case of the weighted projective space, a number of simplifications occur, though these do not come easily; I will try to present the structure of these groups in a way that depends as little as possible on the theory of toric varieties. We assume that $a_0 = 1$.
For notation, let $N$ denote the set $\left\{ 0, 1, \ldots, n \right\}$, and denote by $\Delta^{(k)}$ the set of subsets of $N$ of cardinality $n - k$.
Then we have that $A_k(P)$ is generated as a $\mathbb{Z}$-module by the set
\begin{equation}\left\{ [V(\sigma)] \mid \sigma \in \Delta^{(k)} \right\} \end{equation}
i.e., by the rational equivalence classes of certain subvarieties $V(\sigma)$ of $P$, as $\sigma$ ranges through the $n-k$-element subsets in $N$ (to explain what the $V(\sigma)$ are requires the theory of toric varieties; if one only wants to know the structure of this group, the identities of the $V(\sigma)$ can be suppressed).
Meanwhile, the submodule of relations is generated (as a subgroup of the free group generated over the above set) by the elements:
\begin{equation}\left\{ a_p [V(\tau \cup q)] - a_q [V(\tau \cup p)] \mid \tau \in \Delta^{(k+1)}, p, q \in N \backslash \tau \right\} \end{equation}
i.e., by the combinations described, as $\tau$ ranges through all $n-k-1$-element subsets of $N$ and $p$ and $q$ range, for each such $\tau$, through the elements of the set $N \backslash \tau$. It's difficult to know whether, in general, a more simplified presentation exists, though in the trivial case $a_0 = \cdots = a_n = 1$ we recover the fact $A_k(\mathbb{P}^n) = \mathbb{Z}$.
Finally, the Chow cohomology group $A^k(P)$ is torsion free, generated freely as a $\mathbb{Z}$-module by the set:
\begin{equation}\left\{ c \colon \Delta^{(k)} \rightarrow \mathbb{Z} \mid a_p \cdot c(\tau \cup q) = a_q \cdot c(\tau \cup p), \forall \tau \in \Delta^{(k+1)}, p, q \in N \backslash \tau \right\} \end{equation}
of integer-valued functions $c$ on $\Delta^{(k)}$ satisfying the "linearity" property described.
I believe each group $A^k(P)$ above is actually isomorphic to $\mathbb{Z}$, generated in each case by the function $c$ which assigns to each subset $\sigma = \left\{i_1, \ldots, i_{n-k} \right\} \subset \left\{0, \ldots, n\right\}$ the integer $c(\sigma) = a_{i_1} \cdot \ldots \cdot a_{i_{n-k}}$. Thus the situation in ordinary projective space does generalize, in some sense, to the weighted case.
The ring structure of $A^*(P)$ and its action on $A_*(P)$ are complicated; they are described in the paper.
On the other hand, I'm not sure how to make sense of "intersection theory" when it's no longer cycles that are doing the intersecting. What are the meanings of the elements of $A^*(P)$? I would appreciate insight on this. Thanks to all the commenters on the original post.
