Finding subgroups of a free group with a specific index 
How many subgroups with index two are there of a free group on two generators? What are their generators?

All I know is that the subgroups should have $(2 \times 2) + 1 - 2 = 3$ generators. 

I like to approach this sort of problem using graphs. The free group on two generators is the fundamental group of a wedge of two circles $R_2$, which I picture as a red oriented circle and a black oriented circle. A subgroup of index $ k$ corresponds to a covering map $G\to R_2$ of index $k$. $G$ can be pictured as a (Edit: basepointed) $k$-vertex connected graph with red and black oriented edges such that at every vertex there is one incoming and one outgoing edge of each color. In the case $k=2$, it's not hard to write down all such graphs. I count three myself.
