Dance that we often show to each partner, first of all (3-4)
I believe the answer is:
two-step
'dance' is the definition.
(two-step is a kind of dance)
'that we often show to each partner first of all' is the subsidiary indication.
'first of all' indicates taking the first letters.
The first letters of 'that we often show to each partner' is 'twostep'.
(Other definitions for two-step that I've seen before include "Dance, maybe military" , "Sot wept doing ballroom dance" , "Wets opt out of simple ballroom dance" , "Old time dance" , "Wet pots (anag.)" .)