We can represent the process as a movement in a 3 x 3 table as below, where moving right means picking a red card, and moving down means picking a black card.
You only move down or right, and the probability of moving in a certain direction is proportional to the number of cells in that direction.
We start by filling the table with the number of points you get if you stop at each given cell.
$$
\begin{array}{|c|c|c|}
\hline
\phantom{-}0 \phantom{| \frac{2}{3}} & \phantom{-}1 \phantom{| 1} & \phantom{-}2 \phantom{| 1} \\ \hline
-1 \phantom{| \frac{1}{3}} & \phantom{-}0 \phantom{| \frac{1}{2}} & \phantom{-}1 \phantom{| 0} \\ \hline
-2 \phantom{| 0} & -1 \phantom{| 0} & \phantom{-}0 \phantom{|0}\\ \hline
\end{array}
$$
We can now compute the expected return of continuing from each point.
We start from the bottom right (always 0), and proceed upwards and leftwards.
At each state, the expected return of picking one more card is the weighted average between the best result of going down and the best result of going right.
We end up with the following:
$$
\begin{array}{|c|c|c|}
\hline
\phantom{-}0 | \frac{2}{3} & \phantom{-}1 | 1 & \phantom{-}2 | 1 \\ \hline
-1 | \frac{1}{3} & \phantom{-}0 | \frac{1}{2} & \phantom{-}1 | 0 \\ \hline
-2 | 0 & -1 | 0 & \phantom{-}0\\ \hline
\end{array}
$$
The left numbers are the outcome of stopping at each point, as before. The right numbers are the expected outcomes of continuing to play.
In filling this table, your last computation should be for the top left cell. There, if we choose to move, we have $\frac{1}{2}$ of chance of going right, which can give us 1 point in average, and $\frac{1}{2}$ of chance of going down, which can give us $\frac{1}{3}$ (by not stopping).
The expected number of points for an intelligent player in this game is thus $\frac{2}{3}$.
We conclude that the only places you must stop are the cells in the right column (that is, after having picked two red cards). These are the cells where the left number is greater than the right one.
If you start by picking a red card, choosing to continue does not change your expected outcome (you have equal chances of ending up with 0, 1 or 2 points).
In all other cases, you should continue to play.