The Skills section lists a few examples of ways to spend Threat and Advantage for each skill. Also, the usual chart of ways to spend Ad/Th/Tr/Ds doesn't only apply to combat; you can, for example, gain or lose Strain outside of combat.

Strain is a good go-to when you are out of ideas or are wanting to represent tension, exhaustion, or a sudden surge of renewed energy and drive. Other ways I've used Advantage and Threat include:

When the mechanic was trying to jimmy open a hangar door (leading to vacuum) so the party could escape, I had his sizable amount of Threat cause the door to open too soon, a few steps before he had made it back to the ship. One of the other PCs, covering him from the landing ramp, then leaped to catch his arm and drag him to safety. (It was a complication, not a failure=death scenario, but I liked how the PCs jumped to support each other)

When the Marauder was the only one left conscious/mobile after a crash landing, his desperate and creative attempt to patch a hull breach (using nutrient slurry from the ration stockpile!) netted some Advantage, so it not only patched the hole, but did so with minimal loss of atmosphere (because welding a patch would have burned up more air, I told him). That bought more time for the other PCs to recover and for the whole party to take stock of their situation.

The adventures have specific story driven uses for them on certain checks players make, like social checks, Perception rolls, etc. They give a good idea of how you can incorporate them into game play.

I believe everything you said was correct, except that Triumphs and Despairs do NOT cancel each other out. You can have both Triumphs and Despairs for a skill check.

Like Redford said, look over the Skills chapter to get some ideas on how to spend pips for the different skills. Also, check out some of the published adventures. They usually have examples of how pips can be spent for specific skill checks within the adventure. For example, on page 20 of "Under a Black Sun", there's a Computers skills check you can make to help an NPC decrypt a file. Advantages for this check can be spent to get more information from the files, based on the sidebar on the next page. On page 9 of "Long Arm of the Hutt", they have some special uses for pips when making a Knowledge check to discover something about the chitin that's found on the ship. "Beyond the Rim" has a bunch of these sorts of skill checks laid out.

In fact, when creating your own adventures, it's a good idea to plan out certain skill checks that you know will be available for your PCs to perform, complete with how the different pips can be used for that skill check. In a mini adventure that I wrote, the PCs were to speak with an infochant to get information about the adventure. There was a set fee for the information, plus an additional fee for minor related bits of information. In the adventure text, I specified that each success on their Negotiations skill check would lower the fee by X number of credits, while they could spend advantage to get the minor information for free.

Great advice above, and not to get confusing or anything, but I just wanted to point out one thing. The Blast weapon trigger is unique because even if you miss the intended target, if you get enough advantage to trigger the blast effect (3 adv) you can still do damage to the original target and those engaged with him.

Your game, feel free to use them however you want. Some examples:

Advantages

Coordination: you hit the ground and roll to your feet... with your blaster in your hand (2)

Stealth: You are as silent as a mouse... and manage to grab the Wookiee right as he's about to step on a dry twig (cancel Threat at 2:1)

Resilience: you conclude your 9th pint with an epic belch, the Wookiee does not... the cute Gamorrean bar wench seems impressed with your mating call (1)

Skulduggery: you get the ID card... and his wallet (10c x Adv)

Computer: you retrieve the plans... and also notice the roster - Inquisitor Jenkins is here! (1)

Coercion: the Zabrak backs down... "After you take down Nillus, maybe we can do some business?" (3)

Education: The President For Life of this world is a man named Voss... you were at Space-Eaton together - he owes you a favour (4)

Threat

Athletics: the other sprinter pulls ahead... and you pull a muscle (Inflict strain on self)

Deception: the Bith gangster buys your ruse... but now you must marry his daughter (5)

Medicine: you patch him up... and clear up the human's gill rot (Inflict strain on target)

Leadership: they lower their weapons... but you're sure they're just biding their time until they can mutiny you to death (2)

Discipline: you back away... and soil yourself (1)

Xenology: "Yes, that's definitely a Cragmaloid"... unfortunalely 'definitely' is a deadly insult in Cragmoan (2)

Thanks to the Resilience example, I now can't get this out of my head:

CHUG OFF! - optional competitive recreational poison ingestion rules

Each round, participants roll Resilience with a number of purple dice equal to the round number.

• Net Failure means elimination (from the contest and also of a number of stomachs equal to any net Threat generated).
• Net Success means progression to the next round.
• Triumphs indicate you are holding up particularly well, which unnerves your competitors - upgrade the difficulty of the next check for a number of competitors equal to the number of Triumphs rolled (Despair means they spill their drink or the drink of another patron - see expanded bar-brawl rules (or make them up)).
• Net Threat is always applied as Strain.
• The sentient who rolled the most Advantages downed their drink fastest - they earn the encouragement of the non-competing patrons and so get a boost to their next check.

Ranks of the Blooded talent add boosts to your roll as normal and reduces the difficulty of all tests by one (but you still end up as drunk as the other competitors - it just takes longer to hit you).

in my experience, skill monkey is the best resource for new gms to help think up new ways to spend advantage/threat/triumph&despair. each episode focuses on a specific skill with examples of a situation and how those die results can be described cinematically, without resorting to mechanical effects like "you take strain" or "so-and-so gets a boost on their next roll". definitely something to check out if you haven't already.

I can appreciate that coming up with ways to spend Advantage and Threat (and Triumph and Despair) can be tough. After thinking about it, using it in my games and listening to how other people try to spend the results, I've come to the conclusion that these are hard for most gamers to spend because you're not talking about the result itself.

In other games, you succeed or you fail at an action, and the GM tells you what the consequence of failing or succeeding are. Sometimes there are unintended consequences to the action, but nine times out of ten, if you make a Move Silently roll, for instance, you'll overhear the evil minions drop some hints which could be used to your advantage. If you blow the Move Silently roll, then you're flat-footed against the minions. With EotE (and AoR and FaD), players have a say in what some of these side effects are, and more importantly the side effects don't always stack with the success of the roll! So the trick is to not think about the action itself--it's to think about what happens, good or bad, as a side effect of the action. One of the common things I have to guide my players on (and me, if we're being honest) is NOT to use Advantage and Triumphs to help you get the result you were originally aiming for. If you're trying to Negotiate with a merchant and you fail, for instance, Advantages and Triumphs aren't going to get you the price reduction you wanted. You failed at that attempt. What you could do, though, is say that while you have to pay full price or more, the merchant will throw in some basic gear the player needs as a result, or with a Triumph the merchant might look fondly on you for being such a good sport and paying full price that he gives you the number of a friend who can modify that particular light blaster pistol you just bought from him, or he turns out to be a supplier of black market items. The catch is that neither of those results would be a direct effect of getting successes.

I think one of the best ways to get a feel for how to spend Advantages and Threats is to watch the Indiana Jones movies, particularly the fight scenes. In the fights, stuff happens. For instance, in the Tibetian bar fight scene in Raiders of the Lost Ark, you can see threats and despairs being spent when Indy yanks the hot poker out of Toht's hands, only to have it set the drapes on fire. A minion gets killed, only to knock flaming logs out of the fire. Indy gets grappled by a minion, only to have the minion aid him in shooting the Nazi that's going to kill them both.

Alternately, Indy is trying to sneak aboard a plane. He makes his Stealth check, with some serious threat. He climbs onto the plane, the pilot unaware, but attracts a rival-level mechanic's notice. He gets into a fight, and the GM unloads a dark side destiny point on his roll and he gets a Despair. Not only does Indy attract the pilot's notice, but now an even bigger rival-level threat has noticed him. And this guy is strong.

Personally, the most work that I tend to do in game preparation is come up with interesting ways Advantage, Triumph, Threat and Despair can be spent in the encounters.