The rules for this are a bit scattered through the PHB, but here are the rules relevant to your player's situation:
If your player is attempting a shield bash, he must have the Shield Master feat. This allows you to use your bonus action along with an melee weapon attack action to attempt a 5 ft. shove, or knock an enemy prone, using your shield. (PHB, pg.170)
The rules for Shoving a Creature are found on pg.195 of the PHB.
If your player is attempting a charge, they must have the Charger feat, which allows them to use their bonus action to attempt a melee weapon attack or shove along with a Dash action. (PHB, pg.165)
The rules for Breaking up a Move, specifically Moving Between Attacks, are found on pg.190 of the PHB.
If your player is level 1 the only way they could make a charge is if they were a race, such as Human, that allowed a bonus feat at first level. At level one, they also do not have access to Extra Attack. Here is how this scenario could play out:
With the Charger feat
The player has the option to take the Dash action and move towards the goblins and use their bonus action to make melee weapon attack or shove. They may only attack one goblin as they do not have any feature that allows them to attack more than once or more than one target. They have also already spent their bonus action for the turn.
With the Shield Master feat
The player takes the Move action, moves towards the goblins, and takes the Attack action. After their melee attack is resolved, the player may use their bonus action to attempt to shove a goblin or knock them prone. The player may target a new goblin, or the same goblin, with each attack.
Without either feat
The player takes the Move action, moves towards the goblins, and takes their attack action. They may either make a melee weapon attack, or substitute this attack for a shove, using the rules found on page 195. They may only target one of the three goblins as they are only making one attack.
The player also has other options available to them that do not involve attacks, such as Moving Around Other Creatures (PHB, pg.191) or using the Acrobatics skill to jump over the goblin's space. You could also rule the shield is an improvised weapon (or just a weapon if your Paladin proficient) and allow them to attack with it as a melee weapon. As usual, this is 5e; Many things are purposefully left to DM discretion.