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.
I am brand new, I had initially thought about an Innova bag I saw and some others, and a few people spoke highly of the Fade bags. I went to Sun King here in Sarasota, FL on Saturday and looked at a few bags from a few makers & I ended up choosing the Fade crunch box because of the quality construction,features and great price. It's just what I need as a beginner happy to carry The 9 discs that I've chosen as the ones I'll hone my game with (including my 3 wizards in the putter pocket.)