A friend was asking me for suggestions now that her 2.5 year old is starting to get interested in non board books. This is an expanded version of the email I sent her.
The Olivia books by Ian Falconer. There are now books based off the TV show, too. I like the TV show, but haven't read any of the books based on it yet, so I can't recommend those. The originals are all in black, white and red, and the TV show ones are in color, so it's easy to tell them apart. The stories are pretty short, in fact the one we have is a board book, but I still think it's good for older toddlers.
Books by Arnold Lobel, like Frog and Toad, Mouse Tales and Owl at Home. Very small chapter books, with each story being about 10 pages long. We just checked out a collection of his books on CD from the library and I finally get a break from the CD (which is 13 minutes long and all Moira has wanted to listen to for the last 3 weeks) we have of my next recommendation.
Books by Kevin Henkes. We have Julius, the Baby Of the World, although I wouldn't recommend it for someone who is about to or has just become a big sibling. He has a number of books in this world, where everyone is a mouse. My chiropractor has a couple and Moira really likes them. They are mostly very sweet and show the kid mice figuring out how to share and be friends without adult interference.
Curious George by Margaret Rey & HA Rey. From a modern perspective there's a lot of lessons to be concerned about in these books, but the same can be said of Dr Seuss. I was especially amused when we read "Curious George Goes To the Hospital" and all the kids are left in a giant kids ward and none of the parents stay with them. Moira is in love with George right now. Each story is 20-30 pages long, so this is better if your kid has a long attention span.
The Adventures of Little bear by Else Holmelund Minarik and illustrated by Maurice Sendak. These are also small chapter books. Little Bear appears to be living in Victorian times, but Mother Bear is a very gentle and connected parent. I think these are sweet.
Books by Mercer Meyer. Little Critter is another series where if you have or are about to have a baby, you should check them out before you buy them (the characters are porcupines), but Mercer Mayer has some other series books like Boy, Dog, Frog that are very cute and don't have plot lines about how awful babies are.
The I Can Read series (Level 1 or 2), is actually really great for this age. The stories are usually gentle and the language is simplified but there's usually good plot development. Arnold Lobel's books, Boy, Frog, Dog and Little Bear books are all from that series.