If you want a mechanics-first approach, consider comparison to
grapple and
shove maneuvers, and maybe a few of the DMG maneuvers (such as
climb on a creature and
disarm). Also you could consider
hiding to be a similar action since it is also an opposed skill roll that tosses around a bunch of advantage/disadvantage.
Here is my first pass at specific actions:
Frighten
As an action, you can attempt to instill fear into a creature that can sense you, by threatening them with harm worse than what they've already suffered. Contest your Charisma (Intimidation) against the creature's Wisdom (Insight) or Charisma (Intimidation), whichever is better for the target. You may have advantage if your threats seem credible, or disadvantage if they do not, at the DM's discretion. You don't need to speak for this ability to work, although remaining silent may affect your credibility.
If you succeed, the target becomes frightened of you (see appendix A). At the end of each of their turns, they may make a Wisdom saving throw (DC 8 + your Charisma (Intimidation) modifier), ending the condition on themselves on a success. A frightened creature might decide to retreat, surrender, fight harder, change their tactics, or ignore you completely; the DM decides, based on the creature's personality and the circumstances.
Provoke
As an action, you can attempt to taunt a creature into a reckless rage, by hurling insults or interfering with their goals. Contest your Charisma (Persuasion) against the creature's Wisdom (Insight) or Charisma (Persuasion), whichever is better for the target. You may have advantage if you are threatening the creature's bonds or ideals, or disadvantage if the target perceives you as unimportant, at the DM's discretion. You don't need to speak for this ability to work, although remaining silent may affect your credibility.
If you succeed, the target becomes provoked. A provoked creature has disadvantage on attack rolls and ability checks against any creature except you, and can't move further away from you. At the end of each of their turns, they may make a Wisdom saving throw (DC 8 + your Charisma (Persuasion) modifier), ending the condition on themselves on a success. A provoked creature might decide to attack you, change their tactics, attempt to calm down, or ignore you completely; the DM decides, based on the creature's personality and the circumstances.
Distract
As an action, you attempt to trick a creature into looking away, by doing or saying something misleading ("lookout behind you!"). Contest your Charisma (Deception) against the creature's Wisdom (Insight) or Charisma (Deception), whichever is better for the target. You may have advantage if your distraction is very believable, or disadvantage if it's outrageous or if you've attempted to distract the target previously. You don't need to speak for this ability to work, although remaining silent may affect your credibility.
If you succeed, the target becomes distracted, until the end of their next turn. A distracted creature suffers disadvantage on all attack rolls; all attack rolls against them gain advantage; and any creature can attempt to hide from the distracted creature even if the distracted creature can see them. A distracted creature may decide to take a defensive action, attack the wrong target, move to a new location, change their tactics, or ignore you completely; the DM decides based on the creature's personality and circumstances.
(Note that this is all exactly contrary to my previous advice. I don't generally run this way. But I've played in systems that do, such as 3.5 and M&M and Savage Worlds, and they can be quite enjoyable. Savage Worlds, in particular, has a very flexible and satisfying system for Tricks and Tests of Wills.)