An attempted grapple is made by a strength (Athletics) check by the grappler and the target chooses to resist with either a Strength (Athletics) or Dexterity (Acrobatics) check.
In general, grappling is a strength-based skill and it does not give advantage as part of the grappling process. However, as explained above, there are a variety of class features, spells, and other abilities that can grant advantage on grapple checks. Breaking a grapple is done through a variety of ways like escaping, moving out of reach or incapacitating the grappler.