For triadic chords that sound like this one, the easiest thing to do would be to orient yourself using the interval of a 5th (or its inversion, the 4th).
Simply pair up notes until you find two that are a 5th apart. Since you've doubled the F#, you have a 4th and a 5th between it and the B. In your question, you've identified F# as the root, but this is not correct -- it is simply the bass note. The root is the note on the bottom of the 5th interval (B-F#) or the top of the 4th interval (F#-B).
After you've identified your root, simply stack the rest of the notes on top of the root note in ascending order within an octave. You end up with B-D#-F#, or B major. Either you can identify this chord based on your memory or skill in identifying root position chords, or use this process: starting from the root note, find the intervals created between it and the other notes. For triads, you can easily do this by identifying the major or minor scale that matches the other notes in the chord on scale degrees 1, 3, and 5. D# is the major 3rd above B, so this matches scale degrees 1, 3, 5 of the major scale. If the note was D natural instead, it would make a minor 3rd above B, and the resulting scale and chord would be B minor.
Up until the next step, the octaves and doubling of the notes did not matter. Now, we look at which note is in the bass (F#), and in doing so we understand the full name of the chord. If the root note was in the bass, we would say the chord is in root position. If the 3rd (D#) was in the bass, the chord is in 1st inversion. When the 5th is in the bass, as in this example, this is called 2nd inversion.
So, the name of your chord is B major, 2nd inversion. More formally, this is called
6/4 inversion. (See also figured bass.) Less formally, the chord might just be written as
The process above will work for any three different notes that create a major or minor triad. I would start there, and then as you gain more theory knowledge you can branch out into diminished, augmented, and 7th chords. The process is conceptually similar for those: orient yourself, find the root, determine the quality and name, but more experienced musicians figure this out and name chords based heavily on context and sound rather than logically measuring intervals one by one.