I don't know about any traditions similar to the Tuete, but there are several important supplies your daughter will need.
She will need some Finken
to wear inside her classroom. She will also need a Znünitäschli
to carry her snack. And, a Turnsack
for her gym clothes. The children wear gymnastic shoes
for gym class.
Sometimes the class will have a 'Forest Day' (Waldtag) and the kids go out no matter the weather. You will need waterproof trousers, boots and jacket for your child (no umbrellas allowed!) When the kids go into the forest, they carry their snack in a little backpack, instead of in the Znünitäschli. They also need their drink to be in a resealable container (ie. Sigg).
Well, that is all I can remember from my daughter's kindergarten days! I hope you found this helpful. But as for the first day of kindergarten, I don't know of any special Tuete or traditions like that. We just all went in and took pictures of our kids in their new classroom!