Probability that someone will pick a red ball first?

Your proposed solution is exactly correct. Nice work.

To check your answer, you can use the same method to calculate the second player's probability of winning; it ought to be $1-\frac35 =\frac 25$. Let $P_i$ be the probability that the game ends in round $i$; you have calculated $P_1 + P_3 = \frac 35$. Then $$\begin{align} P_2 & = \frac35\cdot \frac 24 & = \frac 3{10}\\ P_4 & = \frac 35\cdot \frac24\cdot\frac 13\cdot \frac22 &= \frac1{10} \end{align} $$

So $P_2 + P_4 = \frac25$ as we expected.