In a word - No. For this question, you would capitalize only proper names. Some examples would be:
If you have a soccer club who's name is "The Second Grade Tigers"; then, second grade would be capitalized because it is part of a proper name.
A musical band may name themselves "The Kindergarten Cut-ups". Again, kindergarten is capitalized because it is part of a proper name.
If you are talking about a school grade as a whole, it is not capitalized, "The children in second grade painted this mural" or "When children enter kindergarten, they begin to learn about the school system".
If you are using the school grade as a destination, "My son enters second grade next year", second grade is not captalized because it is a destination only. "My son is going to the store" is the same usage as in "My son is enters second grade next year".
Hope this clears up the confusion.