What additional difficulties might the inhabitants of a gas giant moon face when attempting to launch spacecraft beyond their moon's orbit, especially beyond the planetary system? I presume that they'd have to deal with the gravity of the gas giant in addition to that of the moon, but what other issues might they face due to the close presence of the gas giant and/or the other moons.
It's kinda hard to tell what problems they face, there are a multitude of potential issues that all depend on many aspects and your question is relatively broad.
The first issue I see is a mathematical one. In the case of the earth, you can launch spacecraft and almost ignore the sun and moon because their gravitational pull is weak compared to earth's. With your moon orbiting a large gas giant, you put your scientists face-to-face with the three-body problem. This problem hasn't exactly been solved, but solutions can be approximated. However, your advantage here is that you probably have a stable orbit of the moon around the gas giant and the spacecraft is too small to affect the moon/planet's orbits, which makes it much easier. On the other hand, other moons of the gas giant can make it a tad harder again.
Next issue is radiation. Gas giants like Jupiter can have massive electromagnetic fields, which trap ions and emit EM radiation of various wavelengths. These create issues for microelectronics, they can cause single event effects such as bit flips/freezes or have the total ionizing dose stack up until the electronic component breaks. This is already an issue with Earth's Van Allen belts, albeit a mostly "solved" problem, and might be a much bigger problem with a gas giants' radiation.
Can't think of any other noteworthy issues on top of my head.
Other than that, you actually have a ton of advantages. The moon would probably be smaller than earth, so escape velocity is easier to reach. Other moons might have advantageous orbits for sling-shot maneuvers. Depending on whether you want to get to the inside or outside of the planet's orbit, you can use gravity assists from the planet, too.
There actually may be quite a few advantages to having a large planet nearby.
It may be possible to, depending on the distance your moon is orbiting the gas giant, to use target the angle and speed of your spacecraft for a gravitational slingshot. This may enable your spacecraft to use less fuel, to gain speed and if calculated correctly allow ready access to other targets.
Depending on the size of your moon (lets say a Deimos sized moon) if the mass is not too large, it may be possible to have alternate launch methods other than rocketry, such as mass drivers or laser launching, using the slingshot method too.
Gas giants due to their gravity also potentially 'clear' their orbits from asteroids and space debris - Jupiter is known as our solar systems 'vacuum cleaner'.
- There is likely a ring system around most gas giants. Due to their large mass they attract a lot of small particle / ice and dust to form their ring systems. Your gas giant may be no different and this may cause increased difficulty in spacecrafts moving through such a ring system.
- The gravitational pull of the gas giant may produce large stresses on your moon - which may make it unstable. The tidal forces may prove difficult for launch facilities on the moon.
- Gas giants are normally (not always) found further from their stars, this may mean that solar powered craft may not have as much solar energy to draw from as when they are closer. However, gas giants do emit large amounts of radiation - it may be possible to use this as a power source given sufficient technological advances.
- Jupiter has 69 moons of which most have elliptical orbits - although your moon may be one amongst many this may make navigational calculations more complex if you are launching craft in the same plane as these moons. There is evidence also of Jupiter and Saturn 'capturing' moons and large asteroids in irregular orbits, you would need to study these orbits to ensure they are incorporated too.
- Jupiters magnetic field is 14 times stronger than Earth's - it is quite possible that yours would be just as strong. Your spacecraft would require significant shielding to protect both inhabitants and on-board computers.