Hi Igor,
What would be some good beginning programs/tasks to learn Go?
Well, naturally the exercises in "Programming in Go".
But more generally, I'd think about writing programs that will bring you face to face with the particular paradigms that are uniquely Go's.
So, maybe take a program you have that makes heavy use of object orientation including inheritance and rewrite it from scratch (don't convert it!) to Go, using Go's inheritance-free approach. Then add a significant feature to the application and see which it is easier to change, the original or the Go version (but allowing for the fact that the original is in a language you're used to and the Go one isn't).
Another idea would be to take a non-concurrent program that has opportunities for concurrency and try doing a concurrent version in Go (and if you have time in its original language).
If you want more specific ideas then I think a good starting point are the Unix classics: grep, find, ls, and so on---the first two could be made concurrent.