1) (most important) The first tone in the first chord is exactly a semitone (minor second..the distance from C to C#), whole tone (second, the distance from C to D), fourth or a fifth up or down from at least one note in the second chord. Or it is simply a tone shared between both chords (no movement).
Ditto for all other tones. You can memorize what a 4th or 5th up or down is easily using the Circle of Fifths if you don't want to calculate it each time.
2) When the second chord's root is not a major/minor second or fourth/fifth apart from the first note of the last chord...the two chords combine to form a larger chord (think of Calvin Harris's "We Found Love", where D# E G# (d# Minor) is played before E G# B (E Major)...and the two chords form D# E G# B (D# Minor 7th)
So the general trick is either to make each note in each chord a semitone, a whole tone, or fourth/fifth away from at least one tone in the chord after it or the same tone.
Look at the below progression
A# C D#
to
C D# G
Notice
1) the C stays the same between both chords: an ideal "no travel" melodic situation
2) the D stays the same
3) the A# in the first chord is a perfect 4th away from the D# in the second
4) the C in the first chord is a perfect fifth from the G
Now for another chord...
From
C D# G
to
G# C D#
1) the D# stays the same between both chords: an ideal "no travel" melodic situation
2) the G and C are a fifth away from each other
3) the G and G# are minor second from each other (very small melodic movement)
There's no guarantee using this method the mood of your music will be upbeat, melancholic... (what ever you are going for)...but it is virtually guaranteed it will sound very musical/beautiful...if not beautiful toward your desired mood (that's where you artistry comes in...more substitutes for that!)