Late Introducing Finger Foods to 10 Month Old

Start with things like the Gerber Puffs (or the generic version), those dissolve in the mouth, so very little risk of choking. Cheerios are a good starter finger food too. Other than that, you really can give them anything that you're eating (with the exception of nuts, peanut butter, or honey). Cut the pieces up very small so he doesn't choke. I would say probably about 1/2 the size of the tip of your finger. If that's too big, make the pieces smaller.
Yes, you should cook veggies first. It doesn't have to be super mushy, but soft.
Babies can gum just about anything! How many teeth they have is absolutely no indication of what they can or can't eat. My girls were both late teethers and were totally on table foods by the time they were 1.
Once he gets the hang of the finger foods, you can make the pieces bigger. As far as how to serve it, just throw it on the high chair tray. It's also a lot of fun to give them a baby spoon to play with while they eat. At first it's just for banging on the tray and making lots of noise, but they eventually realize it's for eating.
OH! I should also mention that anything with a skin (grapes, apples, hot dogs, sausage links etc) should be peeled before you give it to your son.

Experiment and have fun! It's nice when they learn to feed themselves :)

My son wasn't interested in solids at all till after 1 yr. They have to learn how to chew and swallow without choking. Start with baby food. A baby who's been on a bottle or breast usually pushed food out of their mouth with their tongue at first. They have to learn eating solids is different from liquids. When they can manage that, a good first finger food is dry Cheerios. They are small and dissolve easily. A bagel would be too hard and could be a choking hazard. You are going to have to watch carefully at first. That's why pop corn is not recommended till over 3 yrs old (but if you watch carefully, you can give them one popped kernel at a time at 2 yrs old). The baby aisle at the supper market will have other things that are perfect for baby to grab and chew on. Eventually, limp French fries are good. My son use to hold one in each fist and gnaw on them till he thought they were gone and I'd have to show him the ends were still in his hands. Cooked carrot sticks (cooked long enough they will fall apart easily when gummed) are good about the same time they can manage french fries. Just take it slow.

My family's favorites:

Cheerios and Goldfish crackers(of course!)
Toast with cottage cheese spread on top and a little cinnamon (no sugar, cut in bite size pieces)
Any bread, cracker type things -- I did break these in bite-size pieces and only gave one or two at a time because my kids WOULD shove all the pieces in their mouths without chewing or swallowing at all!
Bite-size pieces of french toast (I didn't give my kids sugar or syrup back then, but put some cinnamon in the egg batter to add a little sweetness -- they loved it)
Elbow macaroni
Pieces of scrambled egg
Blueberries, if too big can cut in half
Shredded cheese
Bite-size pieces of grilled cheese sandwich or cheese quesadilla
Any soft fruit, in pieces
Cooked green beans, carrot sticks, potatoes, peas
Gerber has fun "cereal puff" type things in fruit flavors -- I never thought they were all that nutritious, but fun and easy for travelling

I don't believe I introduced meat this early, but when I did start sharing meat from my plate, I gave tiny, shredded-type pieces to start with. Maybe it was just my kids, but they would put WAY too much in their mouth and then get choked on it if I didn't start them very slowly on meat. If you want more info, the book Super Baby Food by Ruth Yaron is a great resource. It can be overwhelming, and you'll want to skip over the parts about making your own baby food since you're getting past that stage. But it has lots of ideas for toddler food as well, including lots of recipes and craft ideas in the back.
Have fun!

You're not late. : ) Since he is just starting out definitely go with bite size pieces (size of your fingernail) and make sure they are soft and break down easily. You can test foods by putting them in your mouth and seeing if you can break it down simply with your tongue and roof of your mouth since that's pretty much what he will be doing at first. If it's difficult for you it will be too difficult for him with his tender gums. Bagels are a bit tough so I would wait on those.

A. you have to use your common sense. yes, you want to give him soft foods that are easily broken up, he is new to chewing so don't assume he will be able to chew it well enough. don't give him raw fruits or veggies yet. foods that don't need much chewing, only gumming on, are safe. bread is fine, yes, cut it into smaller sizes that he can get his hands on, but no need to cut it into tiny pieces. crackers, etc. stuff that will disintegrate by drool and gumming on it. and of course ANY time you give him anything, make sure you are right by him. what you should probably do if you are unsure is ask your pediatrician, or a grandparent or neighbor who knows your baby. (ps, ten months is NOT that late. relax!)

My daughter choked on a little piece of apple - don't give raw fruits - thank God I was right there and got it out of her throat but it took awhile to get it out and I was frantic