It takes 8 minutes for the sun's light to reach earth. If the sun were to instantly disappear, what would happen? I imagine that Earth would fly away on a path tangential to its orbit. What would happen to our heat and light during those first 8 minutes after the sun's disappearance?
Let's assume that every particle in the Sun quantum tunneled far away from us (there's a near 0% chance this will happen, but let's just assume). For the first eight minutes, nothing would change from our point of view. Gravitational waves propagate at lightspeed, so we would continue to orbit the Sun and see it there.
After eight minutes, however, the lights would go out. The Moon will no longer shine upon us (it receives its light from the Sun), so our only source of light will be that of the celestial objects. Mercury and Venus will have seemingly disappeared from our optical telescopes — they will no longer be illuminated by the Sun. It would take another 30 minutes to an hour for the reflected light of Jupiter to shut off, and soon all the objects in the Solar System will no longer be visible to optical telescopes.
According to Ahad (2004), the illumination by the rest of the Milky Way would be approximately 1/300 of that of a Full Moon. So we might be able to see around thanks to the night sky and our own man-made technologies. However, most plants will not be able to undergo photosynthesis and will die.
At this point, we would be traveling tangentially to our orbit around the Sun, as you mention. We would just keep going into the abyss; it's very unlikely the gravity of any object will pull us in. However, things will get quite cold. The Earth would radiate heat away at a moderate rate, and temperatures would be unsustainable in most parts of the world after a year.
Luckily, the Earth has a nice and toasty core that could provide life-sustaining heat in some places (like Yellowstone), so humans that migrate over there would likely be better off.
Since the universal speed limit is the speed of light, for the first 8 minutes nothing would change. Gravity would be the same, our orbit, as you mentioned heat and light would all be the same. Nothing would change until the final signals from the sun hit our planet and ended.