You can ignore what I said about the Common Data Service if you're using Sharepoint instead. I just made a test app using a Sharepoint List of my own and have checked to ensure that the Search command does indeed work with Sharepoint lists for filtering galleries, but it seems when creating an App from template it automatically uses the Filter command so I had to change the formula slightly.
If you are just simply wanting to search through a gallery of items from your sharepoint list using a textbox search feature, and it's not any more complicated than that, then try my suggested formula. I'll give you some steps if it helps...
1. In your gallery, you've probably got various text labels for showing different fields of a record correct? (you haven't specified all your fields in your original question so for the sake of an example I'll assume that includes things like the School Name, the address, the postcode, a contact number etc.). Check each of your labels in the Gallery properties menu and you may see that each of your fields have a strange ID code (e.g. OData_X0011_ab1)... make a note of these codes and which field they are referring to. Your School Name field will just be "Title". For the sake of my example, let's say your fields are as follows...
"Title" = School Name
"OData__x0011_ab1" = School Address
"OData__x0099_zy9" = School Postcode
2. To make a gallery that allows me to search for a school using any one of the above search criteria (and sort it alphabetically by the name of the school), I would change my Gallery Items property to the following statement...
SortByColumns(Search('School Info',TextSearchBox1.Text,"Title","OData__x0011_ab1","OData__x0099_zy9"), "Title", If(SortDescending1, Descending, Ascending))
Powerapps might give a service limitation error but it should still work based on my own test.
If the text box is blank, then all records will show in the gallery. As soon as you type anything into the textbox, it will search for any school where either the School Name (Title) or School Address (OData__x0011_ab1) or School Postcode (OData__x0099_zy9) matches what is being typed. The good thing about doing it this way rather than using Filter(StartsWith) is that you can type in any part of a school name/address/postcode, not just the beginning of it, even without typing the whole word. This makes searching easier for the app users.
Hope this helps.