I've got a group of 5 players that have split broadly into 2 camps. Either, they're actively engaging with the game (LE Warlock and CG Ranger have formed a frenimies pairing and actively engage in character with each other) or they're incredibly passive about their engagement.
There's three players who spend a lot of time disinterested unless there's combat, entirely ignoring the roleplaying aspect of D&D. I have previously put this down to them being new to the game, but it's something I want to actively help them get over.
The first is a brooding rogue type character who doesn't want to engage for character reasons, which I understand but can be really disruptive when they refuse to actively do anything with the others, or are really passive and just follow the group.
The second is a bard who almost exclusively uses their character to force memes into the game. It can be great fun, but they don't have a developed character outside of "I play despacito" or other common meme references.
The third is the one that annoys me the most. The player is experienced, and his character is a dimwitted barbarian archetype. He refuses to get off of his phone, often having to be called several times before he realises it's his turn (even when in combat!) and it slows the pace of the game a lot. I'm not sure how to approach this without talking about it outside of the game and mentioning how annoying/distracting it is for me.
I'd prefer some ways to approach this in-game and allow them to develop their characters and relationships, but I'm also happy to approach it away from the table if that's a better idea.
You have two issues that are related but different.
Let's discuss them in turn.
Remember: You only have control over your DM style. You cannot control the behavior of other players, only change the circumstances they are interacting with.
The hard truth is this: Any solution will require you to change how you run your games.
The New Players
It isn't a surprise that you have some players that are in to a roleplaying aspect of DND, and others that are engaged by combat.
I recommend you review the section of the DMG that discusses player archetypes. Try mapping your players to these archetypes, and consider what in the game does (and does not) engage them. If you need more guidance than the DMG offers, Robin Laws offers a great breakdown of player archetypes in Robin's Laws of Good Gamemastering).
These exercises will help you shape your sessions to be engaging for your players.
You might find that more "gamey" out of combat challenges will engage your combat-oriented players more than a pure roleplaying experience.
For these players, you should meet them where they are and provide them challenges they will enjoy. Character-drama roleplaying is not natural or even exciting for many players, especially new ones. There are many ways to engage with DND.
Over time, as their comfort level increases, they will begin to veer towards roleplaying. This is a natural arc and development for many players.
This goes for both meme-bard and edgelord-rogue. These new players have gravitated toward easy-to-play characters. This is natural for their stage in the hobby.
The barbarian is a more challenging case. This more experienced player should be setting a positive example for the new players.
The advice above (DMG and Robin's Laws) applies, but experienced players have a greater responsibility to discuss with the DM if something isn't working.
I prefer subtle and at-the-table solutions. Unfortunately, it sounds as if this player is causing a negative effect on the whole group. When there are multiple new players, the responsibility on experienced players to carry the group grows.
In this case, a conversational aside with the player is, in my opinion, the most productive way forward. This is best couched in your DM style. Remember, your DM style is the only thing you can personally control.
If the player is making a good-faith effort, then you can try to meet halfway.
If no resolution can be found, this might be a situation where he isn't a good fit for the game.
There are more subtle solutions, such as telling everyone to set their phones into a communal phone-box before the game begins.
Another indirect option is a general (non-accusatory) email to the group about phone usage at the table.
I haven't tried the email option in a DND group, but I've seen a DM use this technique and have used this technique in group management for non DND-related issues. It is a bit passive-aggressive, but it also explicitly sets expectations. Then no one can claim ignorance.
It is very natural for your new players to prefer combat and play one-dimensional characters. People play the game in different ways. As a DM, you can facilitate the game, but you cannot impose your vision.
What you want to do is provide a game that is fun for them. If you and your group have totally different ideas of fun, that is a problem you will need to resolve...
If you are open to providing a game they will enjoy, even if it means adjusting your expectations, then leap in with both feet. Include scenarios that allow (even require) creative roleplaying for them to solve problems. Your group will grow over time.
The phone issue is a separate problem, that can best be resolved with an adult conversation away from the table.
Robin's Laws of Good Gamemastering
My own experience running and playing games in DND, COC, GURPS, and other systems for nearly 20 years