I am the Dungeon Master in a new campaign. Since I made the players start new characters, they were wondering if there was a fast way to gain experience. The players wanted the game to still be legit, but they also wanted the game to get more interesting quickly. Some of them are not experienced players and I am afraid that what I throw at them might kill them.
I know you're not a native English speaker, but in many circles of general nerd culture in America and on the Internet, there is a phrase for what you're doing right now: your spaghetti is falling out of your pockets.
In all seriousness, though, you seem very anxious about your situation and you should take a step back and relax. You say you're good at improv, but it looks like you're already sweating bullets because your party got sidetracked instead of getting to the quest you wanted to take them on.
I'm going to put my direct answer to your question first and some general advice afterwards.
Your players probably won't miss their characters. Why? Well, you generated them, and even wrote up their backstories. These characters were never made by the players playing them, and while they might have grown attached over a few moments, I find it extremely unlikely that they'll shed tears over a party that lasted for four sessions... that they didn't even make themselves. Even if the characters are extremely cool and interesting, that element of personal attachment just isn't there, which allows the players to truly say that they helped create an exciting story with interesting characters. This obviously varies from person to person, but again, I find it very unlikely that your players would have grown very attached in this case.
Go out with a bang. You know that amazing epic encounter you were saving for the climax of the plot arc? Yeah, run it now. You'll have to make some tweaks because you obviously aren't quite there yet, but get them there as fast as possible. Feel free to kill off PCs or even have a TPK at this point; memorable deaths are often much better than "and then they lived happily ever after."
Here's a few things to keep in mind when starting your next campaign:
Your content will come to light eventually, and it will be good. The quests you've designed will always find a way to come forward. Even if your plot arc is entirely ruined by something the players did, you will be able to recycle the content you made but never ended up playing, and I encourage you to do so for your new campaign. The only things that are truly lost are "hard" materials, like NPC stat sheets, etc.
You're in control, and therefore, you set the tone. Sometimes, it is best to take a page out of Gygax's book; after all, this is your campaign, and you put a lot of effort into it. Obviously you shouldn't take the entire preface from the AD&D DM's Guide to heart, but there is a point where a DM should draw a line in order for there to be some kind of structure, assuming you want your campaign to go anywhere. If your players are goofing around and killing NPCs for no reason, or making light of important people in-character, then they should be ICly punished for it; reprimanded for insulting a nobleman, pursued for attacking innocents, etc. It is also very possible to play a serious game in character and laugh until you're blue in the face out of character. This frequently occurs in the Dark Heresy games that I've played and ran.
You had better get used to murdering your darlings. This is a phrase commonly used amongst writers and creative designers everywhere in the U.S. The phrase means that you'll have to scrap ideas frequently, including ones that you really, really liked, so you had better get used to it. The saying is intended for use in the writing, film, video game, and other industries where a publisher or producer oversees your work, constantly telling you what can and can't make it to the final product based on time and expenses. However, it works just as well for when your ideas can't make it to the game because your players did something insane. And, on a related note...
Plan less. I don't know how much effort you're putting into writing everything ahead of time now, but you might want to ease up on that. From what you're telling me about your role-playing experience, it seems like you've been playing in a "safe" and slow environment where you rarely, if ever, have to scrap or re-do material. This happens literally all the time in regular tabletop RPGs, thanks to the insanely unpredictable nature of 4-5 different people working together. It will save you a lot of anguish if you lay out a basic outline of what's going to happen and then add the details once you're sure the players will be arriving there next session, or maybe two sessions later.
It seems like you've learned a lot already OP, which is great, but scrapping a campaign after 4 sessions (and while your players are all enjoying it) is something you should really avoid. If everyone else is having fun, consider either shaking things up a bit and changing your own notes, or coming up with a way to set them back on track, which doesn't always need to feel contrived or railroad-y.
EDIT: Well, now that SevenSidedDie has made that edit to your post, there are a couple of details that I didn't quite catch before, no offense. Since you said you have a month between each session, it seems like you might be over-planning because you have a lot of time between sessions. Heck, you might even consider having more frequent sessions, if you can't stop yourself from overthinking it in the intervening months. If in-person is not an option, use Skype and/or Roll20.
I don't think there is anything official regarding "back-up" characters in any WotC manuals or guides.
In this specific instance, I think you guys will just have to homebrew it, and decide what's best depending on your campaign setting, your DM, and your PC group dynamic. Either way, I'll try my best to answer your questions from experience:
If this is the case, though, how should we handle their experience? Should all backup characters gain experience at an equal rate as the player's main character? Should they level up at the same rate, but keep the minimum experience for their level?
I think that back-up characters couldn't possibly gain experience at an equal rate as PCs, because situationally, I don't think they'd gain the same XP in combat/story scenarios. If they did, then they would be as involved in the conflict as the PCs, which for story telling and role playing reasons could not make sense. But if you don't quite care for technicalities, then I suppose they could-- their XP might not be justly deserved though.
I also think that would lead to PCs taking death for granted in game; although many players take their characters and backstories seriously, many players care about the leveling up part of their characters. By giving them characters that are where they left off, it might create an environment where death isn't taken quite as seriously.
Is there a different way to handle this entirely?
I think that you guys generally have a reasonable idea; if your campaign is mostly set on the seas, I can understand why "back-up" characters who are crew members might make more sense than other more fantastical, Deus-Ex Machina-y reasons.
What I would personally do, as a DM, is perhaps set them at a level lower than their current PC was when they died, or set the new PC's level to be the same as the lowest level in the party.
There is also the idea that if you are on a boat, ship, or any type of seafaring vessel, at some point you will have to dock at a port-- to restock supplies, conduct maintenance, gather info, etc, etc. This could be a reasonable way to "recruit" new crew members to help fill in the gap left by deceased characters (and a reasonable way to introduce new PCs).
If you guys are open to using the playable races introduced by Volo's Guide to Monsters, a new PC might be a Triton, who guards the ocean depths and might want to join your quest, especially if it is set so close to home. A campaign set almost exclusively on the seas lends itself to including Tritons as a playable character race. I do warn against using these races so openly, as more impressionable players might want to be a "cool" new race, and then you'll have a party full of rare races-- though, if you know how to work with it, that could be fun too.
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