We had to skip rice cereal altogether. Tess will suck on fingers of toast with vegemite in the morning. It started out as just sucking on it,now she actually swallows some! I also give her weetbix crushed, with milk.
I pretty much give her a finger of anything I think of...she sucked on some steak the other day and also some pork. Steamed heads of brocoli, cauliflour, and what you are doing thinking to do with the carrot is spot on.
you could try larger cubes of pumpkin and those sorts of vegetables. The other day she sucked a ritz biscuit till soft and then swallowed what she could.
If you have a look at the Gill Rapley website, she has a great down loadable pamphlet to help you get started. She is the queen of Baby Led Solids and has written a whole book on it.
I also give Tess some puree sorts of things such as yoghurt. She will also suck all the juice out of a mandarine segment and swallow the rest.
It took a while to get going so I was and I am still doing a mix of puree and finger food.
You could also try avocado fingers.

HTH...sorry my post is a bit all over the place.