I'll be honest, we grow our own green beans and carrots and they hardly ever make it into the canner because we love them fresh!! If we want to can them to have an extended supply, I look to my fellow Farmers! There's also nothing wrong with keeping an eye out for great produce that comes into stores. You want to can what you enjoy eating - I prefer to taste the produce first before investing time and materials in canning anything up for my pantry. If you can tasteless carrots, they will always be tasteless carrots. Just like making jam, you cannot take a sour or bitter fruit and expect a miracle of sweetness in the end product.
I prefer to pressure can three vegetables every year: Jade Green Beans, Sweet Carrots and Olathe Sweet Corn. Corn has its own blog post here for those interested in canning corn too.
NCHFP Green Beans Process
NCHFP Carrot Process