How do you do a campaign in a spaceship exploring stuff...

How do you do a campaign in a spaceship exploring stuff? You can't have individual characters doing their thing except maybe in space fighters.

Ground teams, boarding actions, etc. Ship battles can't easily make up the majority of your game.

Trade, raiding trade and all that can be fun. Recolecting things in planets, like animals, minerals or alien tech, Diplomacy and politiking, and depending of the PC whims you could play slavers, pirates or whatever takes your fancy.

Firefly managed.

Puzzles, don't just tell the players what they found, let them determine what it is from your description.

>You can't have individual characters doing their thing except maybe in space fighters.
why not? one guy is engineer, one guy is captain, one guy is pilot, one guy is a gunner or marine, one is a medic, each has a role they can do within a space craft. Star Trek is a pretty good example of this, you have a core crew of interesting characters who make up the important roles, and everyone else on the crew is an NPC.

If you're aiming for a larger than 10 person craft, just have each character as the "head" role - head engineer, lead pilot, head gunner, marine sergeant, etc, so that they lead a team of NPCs.

Games like Traveller do the "each player has a part to play in the ship" very well, and usually aimed at smaller ships with 10 or less crew.

In our Traveller campaign we ended up having 3 of the PCs with their own vessels, so they had a small fleet with NPC crews. This is another option, each is the captain of their own vessel, and they all act together in a fleet (or split off and do other things then regroup)

Check out Traveller, it has everything you need to run an exploration space campaign, or indeed any type of space campaign, hell it's made to handle any tech levels you could run a viking or napoleonic or chechenya campaign if you wanted to. It handles melee weapons and gun combat extremely well, and damage is applied to your physical stats, so getting shot will knock you out if you're not wearing armor. This makes things like western duels work too.

Also being in a fighter is probably the last place you want an NPC - if the fighter is destroyed they will likely be killed. Having your ship destroyed is equivalent of a TPK in D&D. you all take damage or are killed, you're all ejected into space and you're now dependent on your potentially damaged space suit, provided you were wearing one - a star trek crew would just be dead because they dont wear suits in combat.

Deal with shit on various planets, space pirates, circumventing military blockades, drama between crew members, being chased by a big bad.

A thing to note about spaceship based RPGs is that space combat is tricky to run. It's the bread and butter of science fiction shows about spaceships, but around the table, with random elements and gameplay having to get involved, two main problems arise:

1. Getting everyone actually involved. A spaceship moves as one tactical unit, so to speak. Combat becomes a little cumbersome and inevitably, some players will find themselves becoming less useful than others. Even assuming you give very player some role which is "relevant" in space combat (e.g. one is the gunner, one is the pilot, one runs EW or whatever), which sounds neat on paper, in practice you will find that quite often what this means is that everyone who isn't in charge of piloting or weapons basically spends their every turn just rolling the same skill check. To many players, this isn't fun, especially compared to the often far more dynamic nature of ground based combat (where every character gets to move around, decide on their tactics and do their stuff).

2. Far worse is the issue of survival. To put it simply, unless you've got another idea in mind, it stands to reason that if the spaceship blows up, everyone dies. Instant TPK, potentially because someone botched their piloting roll. Obviously, this isn't fun - but the only immediate alternative is to ensure that the players never lose space battles and once they've figured this out it all becomes meaningless and boring. Various games over the years have found various solutions for this, mostly involving alternative outcomes in case of loss in space combat (e.g. everyone has to run to the escape pods and end up on the nearest planet, or they're all captured by space pirates or whatever) but the problem with this option is that it runs a high risk of derailing an otherwise straightfoward scenario. Some players like sandbox style games where one weird outcome can change the whole plot, but others find it less attractive.

(cont.) the other obvious problem with the second option is that sometimes it just doesn't feel like it fits the fiction. In the stories which inspire science fiction RPGs, sometimes there ISN'T an alternative outcome at stake. Very often, it really totally is about the enemy wanting to destroy the heroic spaceship. And in that case, you run back into the issue of putting all your PCs in one space basket, and the risk of an "instant TPK" this poses.

Traveller handles space combat extremely well. Everyone is involved because their either have their own vessel, or everyone has their own position on the vessel which actually matters. Engineers will be fixing thins in combat, or preparing to jump, gunners will be gunning, pilots helping line things up or evade fire, etc. With a well written system (and a decent GM), everyone is involved and has something interesting to do that isn't just rolling one check each turn.
Survival is more to do with the GM and his balance. Obviously a GM in a D&D game doesn't want to cause a TPK, he wants the players to think they're heading for a TPK, but pull through in the end. Same with space combat. You want to present a threat, but not actually kill everyone. Once the PCs get to the stage where they have more than one vessel you can start ramping up the difficulty, as one vessel can be destroyed / disabled, and the other can pick up survivors.

The other issue with having your vessel destroyed is its your main source of transport (and income in Traveller - running cargo and passengers gives you a livable profit) so Ideally a GM should be VERY careful with balancing space combats. A new GM should do some playtests to get a feel of how much damage vessels can take and what space combat will be like. Same with personal combat.

From my experience with Traveller space combat it actually takes a LOT of damage to completely destroy a vessel, and usually a vessel will be disabled and vented before it explodes, killing everyone. PCs are more likely to be sucked out into space during explosive decompression than to be killed in an explosion. Having your ship disabled and left for dead is still a big problem, but a good GM will hopefully have a passing vessel within a few days (or months) or just roll random encounters depending on the busy-ness of the area. Perhaps the PCs are picked up by criminals and have a choice to try to take over a pirate vessel or work with them...