Trajectory: What would happen if the sun disappeared one day?
On the first of January, the direction of the sun from Earth is towards Sagittarius, with an ecliptic longitude of 280.  The velocity of the Earth around the sun is perpendicular that that, and towards Virgo, in particular, it is close to the direction of Gamma Virginis (Porrima), with an ecliptic longitude of 280-90 = 190.  The Earth will move in a straight line in that direction.  (Conveniently Jan 1st is very close to perihelion, so elliptic corrections are needed even less than normal)
It won't actually get near to Gamma Virginis because that star is 38 light years away and at the Earth's speed of 30km/s it would take 400,000 years to get close and by then the star will have moved away from its current location.
It wouldn't come close to a gas giant (none are in a position in which they would come anywhere near the trajectory of the Earth, even without checking, they are all on the &quot;wrong&quot; side of the sun)
It would be very unlikely for the Earth to come close to a gas giant.  If it did then the gravity of the giant planet would change the direction of the Earth, it could go in almost any direction.
For another date, the equation is simple enough: find the ecliptic longitude of the sun on that date and subtract 90.  That gives the ecliptic longitude of the Earth's velocity.
You can now use a star map to find the direction that the Earth will travel in.  On January 1st the Earth will travel in a direction toward an ecliptic longitude of 190, but remain in the plane of the ecliptic.  On the map below, the ecliptic is the line through the middle of the map and you can find where 190 is by looking at the numbers at the top and bottom.
Because in the absence of the sun, there would be nothing to cause the Earth to orbit, it would travel in a straight line towards that point.