I have not toured the area but have mountain biked in the Argentinean side of the Andes (Bariloche, San Martin de los Andes, Villa Langostura). This is a common bike touring region in summer. The scenery is very nice and these days food and lodging in Argentina are dirt cheap, specially if you are coming from the EU. The best months to tour the area are December, January and February. A real touring bike would not work for you. You would need a mountainbike, either rigid or hardtail. Also plan to take some outer shell for the evenings as temperatures drop at night. I've never been to the Chilean side but you probably want to go there also during the summer months. If you decide on the Argentinean side you should fly to Buenos Aires and from there to Bariloche or San Martin de los Andes.