When you take the attack action you can use one attack to grapple, shove, or climb onto an enemy.
You can grab a creature which is no more than one size category higher than you. Using at least one free hand, you make an Athletics check contested by the target's Athletics or Acrobatics check. If you succeed, the target is now grappled by you. You can also grapple an enemy to climb on top of it.
Escaping a grapple.
While grappled, a creature can use its action to make an Athletics or Acrobatics check opposed by your Athletics check. If the creature succeeds, it moves to a space next to you.
Moving a grappled creature.
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Using the attack action, you can attempt to shove a creature no more than one size larger than you. You make an Athletics check opposed by its Athletics or Acrobatics check. If you succeed you either move it 5 feet in any direction or knock it prone.
Climbing onto Bigger Creatures
A suitably large creature can be treated as terrain for the purpose of jumping onto its back or clinging to a limb. Once you make any ability checks necessary to get into position and to get onto the bigger creature, the smaller creature uses its action to make an Athletics or Acrobatics check contested by the targets Acrobatics check. If it wins the contest, the smaller creature successfully moves into the target creature's space and clings to his body. While in the target's space, the smaller creature moves with the target and has advantage on attack rolls against it.
The smaller creature can move around within the larger creature's space, treating the space as difficult terrain. The larger creature may or may not be able to attack the smaller creature, depending on the smaller creature's location, this is up to the DM. The larger creature can try to dislodge the creature as an action - shaking it off, scraping it against a wall, grabbing and throwing it off - by making an Athletics check opposed by the smaller creature's Athletics or Acrobatics check.