After you finish building the pet store you need to adopt a new pet for your toddler. Thankfully there is a free one-star Greyhound.
You will need to buy a 'dollhouse' for your toddler (these are the houses about half-way through the Toddler section of the Children's Store).
Next, you need to make some coffee and then make some Rocky Road (which takes 12 hours).
After all this you will need to go to the park and visit the ducks, talk to a statue in the park, have a staring contest with the statue in the park and finally spend 24 hours having a heart-to-heart with the statue.
This should bring you up to 85% complete at this point.
If at ALL possible save your LPs.
During your usual 'Tutorial Quest' you will need to get a cat, which cost 6 LPs (for the cheapest one- the most expensive one is 80 LPs).
I don't know if you will be needing LPs to launch your baby into the Toddler stage.
I will update you as soon as I'm done in the park!