Attempt
Let $X_i(j)$ be the random variable that is $1$ if coupon collected is of type $j$ and $0$ otherwise, where $1 \leq j \leq k$. Let $X= X_1(j)+...X_n(j)$ And I want to find $E(X)$, but this is going to make thi s too complicated since $X_1(j) $ varies from $1$ to $k$.
How can I understand this problem better? I see that once I can understand what they really are asking, I will be able to think better about its solution.
Even if you do find your $\mathbb E[X]$, it won't give you the right answer; what you'll get is (depending on $j$) the expected number of coupons of type $j$ you find. But you don't care about the number of coupons of type $j$, only if it is $0$ or more.
Rather, it makes sense to define, for $1 \le j \le k$, $$ Y_j = \begin{cases}1 & \text{if some coupon had type $j$}, \\ 0 & \text{otherwise.} \end{cases} $$ You should be able to describe the distribution of $Y_j$ exactly in terms of $n$ and $p_j$.
Now $Y = Y_1 + Y_2 + \dots + Y_k$ is the number of distinct types of coupons found, and $\mathbb E[Y]$ is the expected value you want.