How would the g-forces of being on a planet, or even being the planet itself, in orbit of such a star, work?

Your reference frame[1] would negate most effect until you get really close to the speed of light. Like we don't notice that we're orbiting the sun at 66,135 mph [2] or that we're also moving through the galaxy at almost 500,000 mph [3]

  - https://en.wikipedia.org/wiki/Frame_of_reference
  - http://www.wolframalpha.com/input/?i=how+fast+does+the+earth+orbit+the+sun
  - http://www.wolframalpha.com/input/?i=how+fast+is+the+sun+moving

Don't you still not experience any effects even if the whole system was travelling at relativistic speeds (relative to the galaxy).

You would experience time and length dilation of everything outside the star system.

There will be no additional g-forces if there is no acceleration, regardless of velocity. The related concept is that of an inertial reference frame.

After the actual acceleration there wouldn't be any difference to how life on such planets works. What's difficult for me to estimate is the likelihood of surviving the slingshot itself however. Looking at the closest known orbit around Sagittarius A* [1] you get an orbital acceleration of this star system of around 0.2G. I'm not entirely sure about the math, but I'd imagine the slinghot would have to keep an orbital acceleration of significantly less than the planet's own surface gravity, otherwise it would probably strip its atmosphere.

[1] https://en.wikipedia.org/wiki/S2_%28star%29