There are several ways to think about constructing a major scale.
The first method is to use whole steps (W) (Like between C and D on the piano) and Half steps (H) (Like between E and F on the piano). This method is used by much just learning Theory.
(This method has draw backs and can be confusing for people who do not know piano, whole steps refer to note spacing on the piano)
The pattern WWHWWWH is used to describe how you add each note on.
For example if I am building the C major scale, I would start with C as scale degree 1. Scale Degree 2 would be a whole step up, which is D. Scale degree 3 is E another whole step up. Scale degree 4 is F which is a half step up. Continue adding notes until you have 8 scale degrees. this pattern is the same, if you start on a note like Eb as scale degree 1. Scale degree 2 would be F a whole step away.
The better way to think about a major scale is using Major 2nds (M2) and Minor 2nds (m2).
The pattern to build scales with this is MMmMMMm.
This method is superior because it translates to all instruments and is describes how you would write the notes on staff paper.
The process for building goes the same as above. Starting with F# as scale degree 1, scale degree 2 is a major second away, G#, scale degree 3 is another major second away A#, and so on.
Finally, if you understand all of these and want a quick way to write major scales you can think about the number of sharps and flats in the Major key signature. To use this method first write all of the notes out on your staff paper (don't use any accidentals except for your first note), starting with the first note of the major scale (scale degree 1). Then write 7 more notes, each note a second away (they should all be right next to each other). Then figure out what notes are sharp or flat in the major key signature, and apply that to the notes. This is a more advanced method and requires lots more musical theory background. Refer to how to determine key signatures to learn more about this method.