It takes of course a certain speed to escape the gravitational pull of a planet or a star. My question is this: when escaping a solar system or a galaxy isz their an additional gravitational effect to overcome?

No. When you are on the outer edges of the solar system, lets say just passing Pluto, the Sun's gravity is very weak that far out. Yes, the Sun's gravity keeps all the planets "in line" but as long as you do a quick sling shot maneuver, using Neptune's gravity, you'll very easily "pop" right out of the system. Even little Pluto's gravity won't effect you much. Of course this all depends on the type of drive you are using. With conventional chemical rockets, well... you'll be sling shotting all over the place. But with other more "exotic" drives, you don't have to worry about escape velocity or gravity at all.

If I understand your question correctly - you're asking if the escape velocity for a vehicle to launch from earth and escape from earth and the solar system is different than the escape velocity to simply move out of the vicinity of earth without leaving the solar system, correct? The answer is yes - see this attached site:
Escape velocity - Wikipedia, the free encyclopedia
Basically the escape velocity for an object to leave the earth from the earth's surface is 11.2 Km/s, while the escape velocity to depart from the vicinity of the sun, leaving from the orbital position of the earth, is about 42.1 Km/s. Consequently you need much greater initial velocity to get all the way out of the solar system compared to what it takes to just coast out of the vicinity of the earth alone.