What are the worst-case and average-case time complexities of the greedy algorithm for the weighted set cover problem?
Let $n$ be the total number of elements in all sets in $F$, basically your input size. Maintain a priority queue of the remaining sets, prioritized by cost / number of uncovered elements. Every time you cover an element, update the cost of all sets that cover it. Then there are n total updates, so the total time for the algorithm using a priority queue in this way is $O(n \log n)$.
The linear time algorithm for the unweighted case mentioned by Neal Young in a comment is basically the same thing, just using a bucket queue as the priority queue instead of a comparison-based priority queue.
As for average case, you haven't provided enough information to answer that. Average over what input distribution?