Would you capitalize school district if say our district?

You would capitalize "District" in "our District" if you are referring to a specific, named school district. For example:

* "Our District's budget was approved." (Referring to, say, the "Springfield School District")

However, you would not capitalize it if using it generically:

* "Our district has many new students this year."

The key is whether it's a proper noun (specific name) or a common noun (general term).