i've used various sized Innova bags my self depending on how many discs i need for a certain course for the last several years, and love them. the sport keeps growing and more companies are developing bags, with all sorts of designs. exciting times in the world of disc golf these days.
A soft sided cooler will work great as a starter bag. I had a couple laying around the house that I converted. They'll hold 10 - 12 Discs and are a lot cheaper then starter bags. However, I now have out grown the starter bag and have bought Fade tournament bags for Christmas presents for my boys and myself. Now I just have to hold out til Christmas. Then the coolers can go back to carring drinks.