Without violating the action economy? Play a Monk; Way of the Open Hand
TLDR: by using the Furry of Blows, a Way of the Open Hand Monk can do things like hit and knock down an opponent in combat that you want to do; the sweep kick gets translated mechanically as "try to knock an opponent Prone." (Details below in Monk section) With that said ...
How D&D 5e is set up to work
D&D 5e is a turn-based game that vaguely emulates actual combat when combat happens. What it is trying to capture in the six-second combat round is all of the fast and furious action that both sides in the battle are taking. (See more below). It thus has an action economy that will or won't enable some cool stuff to happen within the constraints of the turn-based combat resolution method. Remember: the whole time your character is trying to do cool and deadly stuff to the enemy, they are trying to do the same to your PC. Who gets to go first is deliberately randomized via the initiative mechanic at the start of combat.
Action; Bonus Action; Reaction; Movement; Interact with an Object. (See Chapter 9, Combat, for details). You get to do one of each, although a bonus action and a reaction are usually conditional on some other game feature. (And there are many. The link has an answer to one such case where I laid out the action economy).
If you, the DM, can fit what your players are trying to do into that action framework, so much the better.
Some of the unusual or cool actions during combat can be improvised, but there is an action cost to doing this. (Chapter 7, Using Ability Scores).
Improvising an Action (Basic Rules, p. 72; PHB Ch 9)
Your character can do things not covered by the actions in this chapter, such as breaking down doors, intimidating enemies, sensing weaknesses in magical defenses, or calling for a parley with a foe. The only limits to the actions you can attempt are your imagination and your character’s ability scores. See the descriptions of the ability scores in Chapter 7 for inspiration as you improvise. When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.
A PC can choose to use their action to, rather than attack, do something Athletic (or even Acrobatic). This is typically resolved by using the ability check mechanic: Strength (Athletics). The DM sets a DC for whatever cool thing the character is trying to do, and a d20 roll resolves it. The sky is the limit.
- Swing from that chandelier!
- Knock that table over!
- Dump the boiling water in the cauldron on your enemies!
The DM's task is to figure out how difficult this is (set a DC for the attempt) and whether or not an opposed check/contest is the suitable thing to do. (See Contests in Chapter 7).
- Something to remember as the DM: a variety of gamisms are built into the game such that resource management is a factor. Some examples of this are:
- limited spells slots
- limited class abilities that reset on a short rest
- limited "powers" that represent that only now and again is there an opportunity to have that cinematic / cool thing happen
- Barbarians Rage is not unlimited; the Fighter's Action Surge is a limited resource as well; other examples abound.
All of the above is put into combat resolution in an attempt at game balance.
- Called Shots: Your "shooting someone in the hand" is not "real life" stuff; it's Hollywood / TV / comic book / fictional cinematic cool stuff. Makes for a good scene, and a good story, sure. (In my formal firearms training, I was always taught to aim for center of mass). Called shots do not have a mechanic in this edition of the game but there are some good suggestions on how to work with that idea in this answer. This element got edited out of the question, but I'll leave it here anyway
Why do I suggest Monk?
Caveat: there is usually a cost for doing cool things, such as giving up an attack (as noted above) or in the Monk's case a ki point cost. (Ki points reset on a short or long rest).
When you spend a ki point to set off a Flurry of Blows ...
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
... the Open Hand Technique at level 3 provides these cool options during combat:
- Knock an enemy prone if they fail a Dexterity check
- Get pushed away {or off of a cliff} by failing a Strength saving throw
- Can't take reactions until the end of your next turn {that opponent will have no chance to make an opportunity attacks against you or your allies since an opportunity attack requires a reaction; see action economy above}
Notice that there is a chance for failure in two of the three above cases. The PC tries to do a cool thing, but since the enemy isn't just sanding there it may or may not work. There isn't a video game style "I win" button; there is a chance of failure because the enemy is trying to do you in while you are trying to do in your enemy.
This is combat
Have you ever been in a fight? Boxing? Wrestling? Martial arts? You are working in a dynamic, interactive environment. Unlike Hollywood's "set up" combat scenes, your opponent isn't just standing there waiting for you to get all awesome on them. They are making it difficult for you to achieve your objective during the six second combat round. What's going on in the round is happening more or less at the same time, except that the initiative order makes it easier in the game to take turns in an orderly fashion so that play moves along. D&D is a turn based game that vaguely emulates actual combat: it is not a reality simulator.
More cool combat things done by Monks
Besides using a weapon or unarmed strikes, Monks can:
Stun their opponent at level 5.
Catch a missile and throw it back at the enemy with the Deflect Missiles skill at level 3.
There are die rolls involved; there is no automatic "I win" button.
In summary ...
As an answer to your question that is different than the other answers: to do more cool things in combat, play a Monk using the Way of the Open Hand tradition.