Children need to be 3 years old by September 31 the year they start Prek 3 (the 3 year old preschool program) in the public schools. So there is public school, but not for your child's age. You will apply VIA LOTTERY in February/March (exact date will be on My School DC).
https://www.myschooldc.org/
There are traditional public schools and public charter schools - both of these are funded differently, although both participate in the lottery. You will have an in-boundary school based on where you live and then you can ALSO apply for the other public and charter schools as an out-of-boundary student.
You will want to do research on this starting in November/December - but right now, focus on childcare - center, family childcare or nanny
Frankly, who knows how tours will be handled at public and public charter schools this year? Or at childcare centers? Or whatever because of Covid-19.