If your animal is tame, then you can ride it. In the Ride skill description, it says about riding animals:

When reared from birth, the following animals can easily be ridden by Small or Medium humanoids, depending on the riding animal's size.

This heavily implies to me that you can ride tame animals regardless of what tricks they know through Handle Animal. However, without knowing tricks, the animal will be very difficult to use practically. You will likely need to make ride checks to control your mount in combat if it hasn't been trained in combat, and you will need a DC 25 Handle Animal check to make it do anything that it hasn't been trained to do already.

You can mount a tame lion that knows Come and Stay, but it will be difficult to bring it into combat, and you won't be able to have it walk alongside you without a DC 25 Handle Animal check.

From the PHB p. 195

Shoving a Creature

Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.

The target of your shove must be no more than one size larger than you, and it must be within your reach. You make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.

As long as the target is only one size larger, no one gets advantage or disadvantage.