We started finger foods pretty early, maybe 7 months - mostly as a thing to occupy Ian while we were eating dinner since he ate much earlier. First, cheerios (or Gerber makes some great puffed cereal, wagon wheels, etc.) Now, at nine months, he won't eat pureed food anymore so we go for anything that is relatively soft, and dice it up (he has no teeth). Examples:
carrots
beans
peaches
pears
banana
avocado
potato (even tried a baked french fry and he loved it)
mac and cheese, or other cooked noodles that dice well
Vienna sausages (these gross me out but he loves them - you can get chicken or turkey)
crackers of all sorts