I've read a few comments in a few places that sometimes players and GMs don't know exactly what to do with an Edge of the Empire campaign. An Age of Rebellion game clearly has some assumptions firmly in place. The players are working for the Rebel Alliance, they are primarily working to oppose the Empire, and if player ideas stall out, the GM can have a mission come down from on high and give the campaign a push.
Edge of the Empire, however, is a bit more wide open. While all of the players are fringers that have some kind of Obligation, this is a bit less structured than the core assumption of Age of Rebellion. I think there is also an assumption that with a wide open range of opportunities that the game should be more "sandbox," with the players flying around the galaxy doing what they will, and dealing with Obligations when they come up.
There is nothing wrong with that setup, but it has the same inherit pitfalls of any sandbox campaign, primarily that players can either have no idea what to do next, thus causing the campaign to stall out, or multiple players have ideas on what to do next, and the party is having a hard time choosing between options, thus causing the campaign to stall out. It also assumes that the GM is going to be able to roll with the player decisions and either have appropriate set pieces that can slot into the player's designs, or will be able to do a lot of entertaining ad libbing in their campaign.
But just because the campaign options are wide open, that doesn't mean that the game doesn't support a strong central concept that might narrow down some of the work the GM has to do to run the game, and to create a bit more texture for the PCs to hang their character concepts on.
When looking at Edge of the Empire ideas, I think it helps to look to the Star Wars material for inspiration for starting campaign structures. Without further ado, let's take a look at the question, "where do I start?" when running an Edge of the Empire campaign.
The Bounty Hunter and His Support Team
This idea is central to the Bounty Hunter class in The Old Republic game, but it's actually appeared a few other places across Star Wars media. The basic concept is that you have a bounty hunter, he's got friends that do the things he isn't quite as good at, and he gets steady work from someone that may or may not be his boss, but is clearly the primary source of jobs that the hunter follows up on.
Having a boss or a business manager lets the player relax a bit when it comes to options. The GM can come up with two or three promising bounties, and the players still get to have some agency on how the campaign moves forward, even as the GM has less than an infinite number of options to develop for future adventures.
Having medics, researchers, and tech support on call all make sense for a bounty hunter that is traveling around the galaxy, because they have to be less of a "jack of all trades" and can focus on being able just bring in the bounties.
The "Bounty Hunter and Support Team" campaign also has a built in campaign twist in the form of rival hunters. In fact, the rival hunter is almost always built into the campaign concept when it has appear in Star Wars media.
Examples of the the "Bounty Hunter and Support Team" can be found in The Old Republic (a bounty hunter, his slicer, and some other underworld pickups working for the Mandalorians and participating in the Great Hunt), the old Playstation game Bounty Hunter (Jango Fett and Zam Wesell working for a Toydarian business manager with a rival Mandalorian on his heels), and the Clone Wars television series (both Cad Bane and Boba Fett have a cadre of hunters, slicers, and specialists that work with them, in addition to other hunters).
A Team of Fixers
This is a setup that I'm currently using for my Edge of the Empire game. You have a small time player on the galactic stage that wants to have some fixers, people with a wide range of skills that he can use to do things he needs done to keep his place in the scheme of things.
In this situation, you can reinforce why the players are working for their new employer by having that employer pick them up from a bad situation and/or being the one that provides their starting ship. This provides a tie to the boss for at least a good number of starting sessions, and can give you a automatic character upon which to hitch their Obligations.
In this setup, the player agency is probably going to come from having more open ended resolutions to the jobs they are handed. Unlike the hunter option above, the team of fixers may not get multiple job options, but they may have a lot of room to figure out exactly how to do the job they are given.
Talon Karrde's "A List" team of operatives (Mara Jade, Aves, Ghent, etc.) is an example of this kind of setup, with characters with some disparate backgrounds and skills working for a strong central figure that still lets them go about their missions with a degree of freedom and trust.
Private Survey Team
Another "fringe" campaign that is can bring together a wide range of characters is that of a team working for a small survey company that goes out into Wild Space and the Unknown Regions to survey planets and bring back information to the "civilized" galaxy.
Characters working for a survey company aren't likely to have a lot of say in just randomly wandering the galaxy, but once they get to an individual planet, they can wander to their heart's content on the planet itself. The prime driver of this sort of campaign will to make sure that the planets are varied, with lots of challenges and sub plots that develop on the planets themselves.
Not only is there likely to be dangerous terrain, animals, a natives on a given planet, but there are smugglers that might wish to do business, or even to hide their location from the greater galaxy. There might be stowaways that are attempting to get lost on unknown worlds that cause their own complications. And it's entirely possible that the players might "moonlight" as smugglers in addition to their respectable surveying job.
The Empire cut funding to exploration projects, but individual Moffs are always looking to extend their influence, so taking on Imperial contracts isn't out of the question as well, meaning that the poor, innocent surveyors might run into Rebel opposition as well.
A survey team is going to need navigators, survivalists, and pilots, but once on planet, having some muscle to protect the rest of the team is likely just as important. Slicers may not be in as much demand on frontier worlds, but you never know when you might run into alien technology and computer systems that predate the Empire or even the Old Republic.
Surveyors are mentioned a lot in Star Wars material, but there aren't many well known characters that have followed this career path. Gav and Jori Daragon are examples of explorers in this mold, but considering they got their patron Hutt killed and started the Great Hyperspace War (on accident), maybe your PCs shouldn't use them as examples.
Private Security Firm Employees
Sometimes there seems to be a fine line between bounty hunters and mercenaries, but in general, mercenaries are hired for much more open ended jobs than bounty hunters. Mercs are usually hired to perform an objective (guard this building, retake this city, etc.) rather than being given individual targets to neutralize.
This campaign set up is going to have a lot less player input on individual adventures, but that doesn't mean it has to be a railroad. Characters will still be able to ask around to get information on their opposition. They might push the boundaries of their own contract to be more pro-active and finish a job before it gets really bad.
Twists in this campaign are going to stem from creating threats that are unexpected. Other employees turning traitor, weird local creatures, and unexpected opposition are all ways to keep the campaign fresh. The team may think they are hired to guard a warehouse from thieves, but those thieves end up being members of the Rebel Alliance, for example.
Planets on these campaigns should be varied and provide a lot of flavor. Not only should the opposition be a bit surprising, but it's important for the player's Obligations to make a simple job much more complicated. If the players are guarding something, maybe they lose that "thing," and then need to track it down to salvage their jobs. Maybe the group is hired to take a bandit fortification, and the team has to determine what to do with the slaves taken by the bandits after the fighting is over.
Soldier types are going to get a lot of millage from this kind of campaign, and pilots might be at home if the group has access to vehicles as part of their job. The group's ship is likely to get less of a workout in this type of campaign, but they may be hired to escort VIPs from place to place, and in this case, gunners and mechanics are going to have a lot more to do.
Mandalorians show up a lot in bounty hunter-centric stories, but there are also a lot of mercenary level stories that feature Mandalorians. The Star Wars universe has other mercenary groups, like the Echani, which are "culture base" mercenary groups, but groups like the Aurodium Swords specialized in doing things like guarding VIPs across the galaxy.
In the books there is also a bit of an example of mercenary guard action when it comes to books that have featured Dash Rendar, as he was contracted to head a VIP's security in Shadow Games as was also hired to guard Luke Skywalker for a while in Shadows of the Empire.
This one is a a bit of a wild card, but could be fun. The group in this case is a traveling band of competitors of some sort. They might be racers, gladiators, shockboxers, or whatever else the players might come up with. The GMs can design adventures based on where the next competition is scheduled to take place.
In this kind of campaign, again, the Obligations should really complicate a relatively straightforward competition, and there should be all sorts of strange side jobs that come up, from smugglers to murder mysteries. Just watch Speed Racer sometime.
When it comes to this kind of campaign, a racer or tech dependent competitor will need tech support to keep their equipment in top shape, which allows for a wider range of characters, and medics and bodyguards are never a bad idea for someone that might make his rivals a bit upset with him.
For inspiration from the Star Wars universe, this is actually the setup of the opening story arc of the Droids cartoon. Not only does Boba Fett make an appearance in this storyline, but his involvement is the result of Obligation!