Should you always create unique keys whenever possible?
For example let’s say I have a table with three fields, student ID, first name, last name and the student ID is the primary key.
If no two students have the first & last name, should I create a unique key for those two fields?
Yes, you should use unique indexes even when you already have a primary key when the column or combination of columns are unique. It’s good to have constraints in your database to prevent bad data. However, this is not what you have in your case. Even if you currently have no students with duplicate names that can easily happen in the future. Names are not unique in the world.
U.S. Social Security numbers are almost always unique (they can be reused after a number of years, but it’s unlikely to ever happen in your case), so they might make for a good candidate for a unique index. If you have non-U.S. students though then you would need to make the column nullable.