When we first introduced cutting to my son at 15 months, it was with a plastic dough cutter. The dough cutter + scraper actually happens to cut bananas perfectly! It was not too sharp, either. I modelled cutting a banana slowly and offered it to him to try. He needed a bit of help with how to hold it and push down, but we managed a few slices together.
We also had a wavy crinkle cutter but didn't end up using it that much. From the age of 2 my son has been using these Kuhn Rikon knives in the kitchen. They are great because they cut well without being very sharp - tricky to explain - they remind me of a kitchen palette knife in terms of sharpness.
You can get just the 2 knives together, which includes a straight blade and a serrated blade (also I love the sweet dog design on them). We bought the set of 3 from Kuhn Rikon and they included 2 dog knives and a pair of duck snippers, for things like herbs. We have been thrilled with them and my son has enjoyed using them. We even got a set each for the grandparents' houses!
With things like carrots, my son enjoys peeling them and sometimes needs a hand to start cutting the thicker parts, then he happily cuts them into pieces. We have been including him in vegetable chopping from when he was really small, so that he can be a part of the family and help us cook.
He stands on our beloved learning tower to work on the kitchen countertops and other times he will be at his weaning table and chair. I recently found a perfect small sized bamboo chopping board from Tiger which works very well with preparing food. I end up using it in the day for cutting things like apples and lemon, too, so it's much used by everyone in our household.
