I don't think there is a "set" age for starting finger foods...they should have a good pincer grasp working to pick it up (I've heard). I tried puffs about 3 weeks ago and it didn't go over too well, but will try them again soon and will move onto well-boiled diced veggies/fruits from there. I am not trying the teething biscuits because they can break off and become a choking hazard.
We just started puffs yesterday but she has been feeding herself for about the last 6 weeks. We started with sweet potato fries. She also does green beans, brocoli, carrots, butternut squash, and zucchini. I just make sure everything is well cooked to a very soft consistency, and kind of in a stick shape so there was something to grasp and then something sticking out of her hand to eat. We reintroduced the spoon for some cereal and it's going better than it had been before but DD really prefers to feed herself.
When they can sit unassisted finger foods are okay. DD doesn't have a great pincer so we haven't done smaller foods yet, we serve everything to her in fingers or strips she can hold in her fist and bite off of. Pretty much any fruit/veggie/meat can be prepared in that way, she's a great eater. She also loves toast, egg yolks, hummus, and rice cakes.
Re: At what age? Re: finger foods like Cheerios
This.