From your point of view it would be instant. Sun is there, then poof! The sun is gone and so is its influence. We only know this event happened ~8.5 minutes before we saw it because we're so clever :-). However there's no way to detect it early and warn ourselves because no information could reach us any faster than the sun's extinguished light and missing mass/gravity wave.
With our liberation from the sun’s gravity, we’d be traveling at the same speed as before – about 18 miles, or 30 kilometers per second. Instead of curving around the sun we would continue straight out into space. So Earth would be traveling at the same speed as always into eternal night. We would loose some tidal influence that could cause some tidal wave surges around the planet.
If you were on Earth’s night side when the sun disappeared, you might not notice anything. But there might be clues in the night sky. For example, if there were a full moon – which shines with reflected sunlight – its light would disappear. Over the course of several hours, the planets would wink out one by one, as they reflected the last of the sun’s light to us.
I know you didn't ask this, but more disturbing of course would be our lack of a sun for a heater and for photosynthesis. "Within a week, the average global surface temperature would drop below 0°F. In a year, it would dip to –100°F. The top layers of the oceans would freeze over, but in an apocalyptic irony, that ice would insulate the deep water below and prevent the oceans from freezing solid for hundreds of thousands of years. Millions of years after that, our planet would reach a stable –400°, the temperature at which the heat radiating from the planet's core would equal the heat that the Earth radiates into space," explains David Stevenson, a professor of planetary science at the California Institute of Technology.
Although some microorganisms living in the Earth's crust would survive, the majority of life would enjoy only a brief post-sun existence. Photosynthesis would halt immediately, and most plants would die in a few weeks. Large trees, however, could survive for several decades (or until they froze to death), thanks to slow metabolism and substantial sugar stores. With the food chain's bottom tier knocked out, most animals would die off quickly, but scavengers picking over the dead remains could last until the cold killed them.
"Humans could live in submarines in the deepest and warmest parts of the ocean, but a more attractive option might be nuclear- or geothermal-powered habitats. One good place to camp out: Iceland. The island nation already heats 87 percent of its homes using geothermal energy, and," says astronomy professor Eric Blackman of the University of Rochester, "people could continue harnessing volcanic heat for hundreds of years." The technology would have to be improved for a -100°F or more temperature differential. And their source of food would be extremely slim with the rapid loss of plants and few sources for critical vitamins.