An idea: draw two circles with radius 1 and center resp. around the two given points, and now "shrink" the exterior of both circles to the point (0,0) and ALSO the interior of both circles shrink to the same point (0,0) ==> you get the plane minus those two points is homotopically equivalent to a bouquet of two circles (i.e., the figure ), and it is well known, and as far as I remember not hard at all, to prove that this bouquet's fund. group is the free group in 2 generators.

Tonio