I think you need to prepare yourself for the fact that she will gag - which someone has already said, is different to choking. Gagging is the bodies way of saying, that's in the wrong place - it's all a part of her learning how to handle solid food. I don't know that I would go for carrot, but perhaps really well cooked potato or pumpkin as they will disintergrate fairly quickly in the mouth. Unless she's appearing upset by the process, continue with it. Babies will gag but they usually don't become upset by this, as I already said, it's just a body reflex.
Like others said, just stay close so you are handy incase she does get something 'stuck'.