A monster group consists of three kobolds.

Firstly, as DM do I roll initiative once for the whole group?

Secondly, when it is the kobolds turn, do I roll a separate attack for each kobold? If so can they attack different characters? So does each kobold get an attack every turn?

1. Should you roll initiative separately or together?

Making just one initiative roll for a whole group of monsters is normal play:

Initiative

Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time. (PHB, p. 189)

Making one roll for a group is often advisable, especially for new DM's as it both speeds up and simplifies combat.

(However, in large groups this can sometimes cause issues.)

2. Always roll the monsters attacks separately.

According to the Order of Combat:

During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. (PHB, p. 189)

So, every kobold will take a turn each round. And, they could all choose to use their action that turn to attack - so they could all attack each round.

However, there's no guarantee that they would do so, as there are plenty of ways an NPC could spend their action other than attacking (Dashing or Helping, for instance)

If they do attack, they can definitely attack different people. If only one kobold took an attack each round then you'd really only be fighting one kobold at a time three times in a row (a much easier fight), rather than three kobolds at once.

When multiple creatures are attacking, even with the same target, all attacks should be rolled separately (though the separate dice could be rolled simultaneously). This prevents damage from being too swingy.

If either everyone hits or everyone misses in any given combat round then it becomes harder for players to gauge how hard a fight really is, and more difficult to prevent PCs from being knocked out through careful healing.

Rolling the initiative is debatable, but in my experience (from my own group and podcasts such as critical role and the adventure zone), the majority of people roll by group, so if you have 3 kobolds and 2 orcs you roll 1 initiative for the kobolds and 1 for the orcs to save time.

For your second question the answer is yes, each kobold gets their own turn to move and attack, whether you decide to roll by group as above or give each their own initiative. Each kobold can do whatever they like on their turn including attack several characters.

As dm/gm though it is your job to be fair when deciding what they do. Are the kobolds cowardly? If so they may try and gang up on a weak enemy to finish them off or may attack from a distance. Has the bard been targeting one with their spells or insulting them? Then that kobold might be mad and decide to attack the bard specifically.

NPC initiative varies by DM, but each enemy gets an attack

I'll answer the second part of your question first since it is easier.

Each enemy gets a turn every round.

Regardless of how you roll initiative when a creatures initiative comes up it gets it's turn. On its turn it will get movement, an action and a bonus action just like the PCs. How it can use these actions are determined by the creatures stat-block. Typically in combat they will take the Attack action as their main action, but they may also have special abilities or tactics to use instead. It is up to you, the DM, to decide how they use their turn.

Each creature acts independently of its allies but can work together with them. Intelligent creatures will often use their movement to flank and gang up on an ally. Or they can all rush to engage different opponents, spreading the parties resources. Different kinds of creatures will have different behaviours and strategies in combat and you should try to play this out at the table.

Group initiative is an optional rule

Rolling initiative for a group of enemies together often helps to save time, and the DMG recommends it for identical creatures, however it can lead to combat that swings wildly from one side to the other. There are times when it is a good idea, and times when it is better to roll separately. For a good break-down of how to run creatures in group combat see: How can I solve burst damage problem when rolling group initiative?

I only use group initiative when there are multiple enemies types or a large number of enemies. If the number of enemies is roughly equal to the size of the party I prefer to role individually. I find this evens out damage in combat and allows the players to better plan their attacks.

When using group initiative I prefer to role initiative by groups of no more than 4. I break my enemies up by creature type, then into melee and ranged fighters. This helps to prevent swingy combat and keep the flow going.

In your situation I would roll individually for each kobold at this level. At later levels when there is a Dragonborn spellcaster leading his kobold minions I would roll the kobolds together, but separately to the boss.