Letters
Depending on your high school, you may be required to ask your teachers to write your letters of recommendation at the end of your junior year, so May-ish.
Essays
The Common App resets August 1, so I would start writing your Common App essay a bit before then. Write any supplements after. You don't need to finalize your school list yet...
your list
Create your list of schools between August 1 and when you start your senior year. Try and get as much as you can done before you start. Choose schools that have strong STEM programs that are well funded.
deadlines
Please, please, please keep your deadlines organized. If you're applying to schools that don't use the Common App, deadlines may be different, and you don't want to miss out on applying to a school you really like.