To go to Nursing School, you must have high smarts, and go to college and study nursing.
That said, start by creating a character until you have decent smarts of at least 40+. This number will be fine enough, as you can increase this all the way to 100 with the next step.
After you create your character with 40 smarts, you will then need to age until the option to read books and go to the library unlocks. When all of this unlocks, you will be able to read books and go to the library under the activities tab. A combination of both will be enough to raise your smarts stat to 100.
When you get high smarts, you will then need to age to the point where you finish High School. Once you finish high school, move on to college and choose nursing as your major.
After this, finish college by aging a few years, and you will be able to go to Nursing School by selecting the seek higher education option. If you don't see this tab, you can always find the option to go to Nursing School under the education tab in careers.
That said being said, once you completed both nursing school and college, you will be able to become a nurse by finding the job under the career tab.