Best and easy-to-master free softwares for drawing geometric figures in math papers?

I would also recommend GeoGebra, it can also generate code to use with the TikZ or PSTricks packages, so you get high quality images.

If I want an illustrative graphic, TikZ would possibly be the system of choice for a LaTeX document for me.

If you want to actually plot different curves, I would support Daniel's suggestion of Geobgebra which is possibly the easiest to use free package out there.

If you have data to plot, I would suggest spending the time to learn R http://www.r-project.org/ which can create pd pdf plots (amongst other formats) to embed in a document.

(If you are in need of computation, Octave and Maxima may be of interest, but if you need simple graphs, they will be a lot of work for you.)

As always (and with the risk of sounding like a parrot) I would like to suggest Inkscape which with some meddling also can export PDF's that you run through LaTeX and get the best of two worlds.