When you perform an interplanetary voyage, it saves you a lot of fuel when you wait for the planets to be aligned in a way that you can reach your destination with a simple hohmann transfer. In the case of Kerbin->Eve, that would be when Eve is about 54° behind Kerbin (there is a handy online alignment calculator for KSP).
When you enter the Eve planetary system, you should first get into an Eve parking orbit.
You can do so with minimal fuel when you adjust your course early so you get into Eve's atmosphere at a height of 72km. That way you can perform an aerobrake maneuver and get into an eve orbit with minimal fuel expense. Aerobrake at Eve is now much harder - you would surely burn in atmosphere, unless you will bring heat shield - but then you don't need it on Gilly.
Then you just need to perform a normal Hohmann transfer from Eve to Gilly, like you would from Kerbin to Mun (although you need to aim a lot better because Gilly and its gravitational sphere of influence is tiny). Because Gilly has such a low gravity, landing on Gilly is rather trivial (It does in fact require so few energy that a Kerbonaut on EVA can get from a Gilly orbit to the surface and back to the ship using just his jetpack. But this stunt is rather tricky to pull off, so you should rather land with the capsule).
When you got your science and would like to get back, it is a lot more effective to return to a low Eve orbit and start from there than starting directly from Gilly. This is because of the Oberth effect: expending fuel near a large gravity well is more efficient than in a high orbit around it. Again, you should have some patience and wait until the planetary alignment is correct (we return to our handy calculator and find out that this is when Eve is 36° behind Kerbin).
By the way: As you might have noticed, a mission to Gilly is a mission to Eve orbit plus a field-trip to Gilly. That means you should try to combine it with some exploration of Eve. Returning from the surface of Eve is by far the most challenging thing you can do in KSP. When you want science points, there are many more lower-hanging fruits for you to pick up. But just landing on Eve when you don't intend to get back is quite simple. A simple probe-lander to Eve surface just needs a probe-core, a parachute, a heat shield, an antenna, a few batteries and solar cells and the scientific instruments. With just about 1/2 ton of additional payload you can get surprisingly valuable instrument results from Eves upper atmosphere, lower atmosphere and surface.
When you have no scruple to sacrifice a Kerbal for science, you could expend about one ton and use a manned no-return Eve lander for crew reports, EVA report and a surface sample (you monster).