If you are interested in the concept of feeding finger foods look into something called Baby Led Weaning. I did it with my son and we started at 6 months (we did cereals and purees the month before that) and he has never had any issues. He's always been very very interested in any foods we eat and that had an effect on our decision. He eats things like large strawberries with the greens cut off, graham crackers, peeled apple slices, large florets of broccoli, cauliflower, boiled/baked potato wedges (red/russet/sweet potatoes), steamed carrot sticks. You pretty much want things that is the right size/shape for them to pick up and hold in their hand but not small enough that they can get in their mouth whole. We still chopped things up and gave things like steamed peas to let him work on his pincher grasp. I do recommend reading up on it a bit before you start so you get all the necessary info and you can make the right decision for you guys. Babies are perfectly able to eat whole foods without teeth, their gums are really really good at mashing things, its just knowing which foods are ideal. I didn't do cheerios until about 8 months because they don't dissolve all that easily. I usually try anything before he gets it that way I know if it dissolves fast/requires some chewing so I know what to expect in his reaction. Also, when you do go for cheerios get the regular ones not multigrain (lots of sugar in the multigrain ones) and with puffs all the store brands are an A shape that's harder for them to pick up so go for gerber or Happy baby (these look/taste like cheerios) for puffs.
You'll still want to keep an eye out for allergies the same way with baby led weaning as you would with the "normal" feeding routine. Know what to do just in case your little one ever does happen to choke. Also be aware that they do have a shallow gag reflex and they will gag stuff back up towards the front of their mouth more often than not if they can't chew/swallow it. You'll get used to what they do when they have that gag reflex triggered and learn that its really not a panic situation. If you're interested in it DO look into it though, look it up online, go to the library and rent books about it! My son's pedi seems to think its not a bad idea though he doesn't have experience with it and my son has always enjoyed it.