Recommended and mentioned products
Cytoscape Official Web Site
TikZiT is a graphical tool for rapidly creating graphs and string diagrams using PGF/TikZ.
I would recommend doing graphs in latex using tikz. There's a bunch of editors (some online too) that slow you to create them easily like this one: https://tikzit.github.io/.
Gephi is an open-source software for visualizing and analyzing large networks graphs.
I was a big fan of gephi. It hasn't seen active development for going on 4 years. However, if you just want to use it for learning interactively it should still be fine for your purposes.