Frances the Badger series, by Russell Hoban

Russell Hoban's Frances stories are written at approximately the same reading level as Frog and Toad, but while Frog and Toad titles include a handful of short episodes, each Frances story takes up a whole book.

Frances stories are a little didactic, each involving a particular childhood problem such as going to bed, learning to accept a new sibling, or celebrating other people's birthdays without sulking that it isn't your own.  My favourite story is Bread and Jam for Frances which is the least moralistic of the bunch. 

Frances herself is charismatic and true-to-life.  She invents silly songs throughout the day, which I quite adore.  They define her character, and remind me of my own daily childhood compositions, or the way my three year old nephew revels in wordplay.  She acts intermittently stubborn, jealous, selfish, mopey, confused, happy, and loving just like any other preschooler or kindergarten aged child.

Titles:
Bread and Jam for Frances
A Baby Sister for Frances
Bedtime for Frances
A Birthday for Frances
Best Friends for Frances
A Bargain for Frances