Wide clustered index vs multiple narrow nonclustered indexes? Say I have a contrived Student table like so:

CREATE TABLE Student (
    Id IDENTITY INT,
    SchoolId INT NOT NULL,
    FirstName VARCHAR(20) NOT NULL,
    LastName VARCHAR(20) NOT NULL
)


Instinctively, I'd make Id the Primary Key (and thus the clustered index). However, I'd find myself searching by SchoolId so I'd make a nonclustered index on SchoolId.

How would this fare against having the Primary Key (and clustered index) to be SchoolId, Id? I will always have the SchoolId if I need to search by Id, so I'll get to use the clustered index anyways, and if I need to search by SchoolId only, the records will be physically next to each other. 

If I were to do any type of searching or batch updating, they'd be on SchoolId specific records, e.g. find all kids with name/number/whatever at SchoolId. I'd never do these types of operations across multiple SchoolIds in the same transaction. Does the benefit of having these records physically next to each other make this method much better than simply having a clustered index on Id?

Are there massive downsides to using the latter? I'm still new at this and there's plenty of topics I don't fully comprehend yet (e.g. fragmentation) and how it would factor into a situation like this.
If you will always have SchoolId then you could benefit from making the clustered index a composite key of SchoolId, Id as you won't have to have an additional index on SchoolId to avoid a table scan.  Not having the additional index will let Inserts/Updates/Deletes complete faster as the transaction only has to update one index.

You may find as you create other queries that use the other fields in the where clause that an additional index on those might be beneficial so SQL Server can seek right to that index leaf which will help reduce the number of row returned.