SEO is about understanding what people are searching for when they're looking for schools online. The words people type into a search engine are called "keywords." So the first step in optimizing your website for SEO is to determine what keywords you want to optimize for.
You may have a few keywords in mind that you would like to rank for such as your school type and location. This is a great place to start! You can enter those keywords into a keyword research tool and discover average monthly search volume, similar keywords, and which variations of your keywords are most popular among searchers.
Type "private" and "school" into a keyword research tool and you may discover highly relevant, highly searched for related terms such as:
Volume - how many people are searching for a specific keyword? The more people are searching for a keyword, the larger the audience you could reach.
Relevance - if a term is frequently searched for but it's not relevant to your website, utilizing this keyword will not help your website optimization.
Competition - the higher the search volume for a keyword, the greater the competition to rank higher for this keyword.
You do want to target terms with high search volume that your audience is searching for but it may also be advantageous to target some terms with lower search volume because they're far less competitive.
Once you know your keywords, you can begin optimizing your pages utilizing these keywords within title tags, meta descriptions, body content, alt tags and URLs.