[RPG] Do D&D 5E characters get the characteristics of both Classes AND Backgrounds

If you want to be effective in skills and combat, I think you could go straight bard. However, I think it could be extremely effective to take a 1-level dip in Warlock with the Hexblade Patron (in Xanathar's Guide to Everything), and do the rest of your levels in Bard. I would take the dip in your first or second level, but then I would take it straight to Bard 6 by 7th level for Magical Secrets before considering any other dips.

The 1st level of Hexblade will give you:

• 2 Warlock Cantrips
  • I recommend Eldritch Blast + 1 other
  • If you can use multiple books, you should consider 1 of the cantrips in Sword Coast Adventurer Guide - I would go Booming Blade for flavor (note that Bards cannot get these by RAW, but they are on the Warlock spell list)
• 2 1st level Warlock spells that you can use your Warlock slots for (these do not stack with Bard slots or spellcasting) or your Bard slots
  • I recommend Hex + 1 other
• Hex Warrior
  • Proficiency with Medium Armor, Shields, and Martial Weapons
  • Pick a melee 1-handed melee weapon and use Charisma to attack!
  • Now you can max Charisma, shoot for a 14 in Dexterity with medium armor and a shield and pull off a 17 or 18 AC and put whatever other stats in Constitution for HP or Wisdom/Intelligence to improve skills.

This will make you effective in combat. You already get a lot of skills as a Lore Bard:

• 3 skills at Bard 1
• 2 skills from background
• 3 skills at Bard 3
• possibly 1 or 2 from Race
• expertise in 2 at Bard 3
• expertise in 2 more at Bard 10

This will help you be about as effective in combat as a Valor Bard while still getting the skills of a Lore Bard.

The thing done wrong was not accepting the re-roll offer

If the offer is still open, take the sorcerer and wizard players up on their offers to reroll characters if you want this game to go forward.

Why should I do this?

It appears that you were deliberately trying to create some intra-party friction (similar to how Adventurer's League uses Factions for organized play) and that this one mission was bound to create friction based on how you had set it up. Why you needed to get the friction to come to the surface early is unclear, but that's the side effect of the faction quests that you presented.

The other thing done wrong was not perceiving an expectations mismatch.

While I personally like what you were attempting to do insofar as role play is concerned, getting a party to work together with opposed motives is a tricky thing to pull off in this particular game. I've rarely seen it done well (it can be done). I've more often seen it kill off a party and a table.

One of the things you are working against whenever you try to do this is the fundamental structure of D&D 5e as a game. It's built as a team game where a group of people with different skills and talents together overcome challenges, solve puzzles, discover things, and take on adventures.

Most groups of PCs go through four phases: these phases of making an effective team are forming, storming, norming and performing. (Small group dynamics, 101) Your two brothers and three others set up is a classic example of this.

By your having inserted deliberate friction into the team this early on in the campaign, your team of PC's never got past the forming and storming phases. The disagreement on this mission was a reasonable prediction. The offer of "we'll roll up a new character" is as good a solution as you'll get if you want this to move forward.

That your players offered you this solution is a bright shining green traffic light: they recognize the expectations mismatch, and they are offering you a way to keep this campaign alive. Go with their suggestion!

Recommendation

Take them up on their offers! (That wizard and the sorcerer). Have them roll up two new characters so that this party can continue on in their adventures in your world. As DM, put in the work to help them fold their back story into your world so that it aligns with the general theme you already have, and that is mostly acceptable to your players so far.

What could I have done differently?

1. Not create irresolvable character motives in the first place.

   The tension and conflict that you had within the campaign structure may work in writing a book, or a movie or a video game, where the author controls all characters and the narrative. It does not translate well to tabletop RPGs where other people control the protagonists (player characters). The sorcerer was, from your description, more or less maneuvered into a conflict with his new adventuring partners by authorial structural decisions.

   From a design perspective, the mission your sorcerer had would have been better assigned to an NPC whom the players meet/ally with due to something else. This would free up the team to accept, or not accept, participation as a team.

2. The sorcerer's initial mission needed to be more attuned to the general team building that early adventures are meant to establish, rather than in direct conflict with team player goals.

   You mentioned in a comment that you were trying to pull off what Matt Mercer does (apparently) effortlessly. What Matt Mercer and his experienced group of players can pull off doesn't happen overnight. He's been at this for a while, and I'll also point out that the show Critical Role is first and foremost a show meant to entertain. It's not a primer on "How to DM 101." (Though I'd love to play at his table).

3. As a new DM, it's simpler for you to begin a campaign with the basic structure in mind of "a team doing things" and as your players' team forms, storms, norms, and performs, you later introduce the topic of possible differences in player character goals. Crawl, walk, jog, trot, sprint.

4. What you can try to do differently is make sure that you get all of your players to buy into what level of interpersonal tension or friction they can accept before you embark on the first adventure. Unless you get buy in from the players, the "I didn't sign up for this" reaction is predictable.