I'd def recommend watching Winter Soldier before watching Agents of Shield. They start dealing with the aftermath of the movie in season 2. It's an alright show, it's getting better but you have to power through season 1. I haven't seen any of Agent Carter yet.
The aftermath was chronicled during season 1, starting with the episode titled "Turn, Turn, Turn". The second crossover with Captain America: The Winter Soldier is called "Providence", and it follows the episode above in airing-date. You can watch the show without seeing The Winter Soldier, as the aftermath of the movie is alluded to without going into specifics, though every show after the former deals with S.H.I.E.L.D. finding out about _____, and dealing with those consequences.
The first season is hollow and formulaic, preferring to feature a single bad guy that is ousted by the end of the episode until around the crossover, and then it begins to explore mysteries-in-canon and the people pulling the strings. Season 2 is pretty stellar. The mid-season finale of S.H.I.E.L.D. is a direct reflection and the beginning of one of the movie universe's future films, and explains subtly how characters like Scarlet Witch and Quiksilver (Avengers: Age of Ultron) have "miraculous powers".