I'm going to be developing a android game that will be puzzle based so will have a game board with different pieces. I don't have a lot of experience designing graphics for apps so I was wondering which would be better for developing the game board and various icons and pieces in the game, Inkscape or Gimp?
I'd use InkScape as your base, and convert to raster. 9-patch versus BitMap is more a matter of what you're trying to do with the raster image.
The advantage of InkScape is that you can start with an image and then scale it reasonably well. Vector graphics are good for scaling.
You don't need GIMP, unless you want to scale a raster to a raster. However, I'd avoid scaling rasters unless you absolutely have to. A vector scaling does much better. Always keep the vector as your "master" image, then convert to .png to get a result you can use in Android. It's the graphics equivalent of compilation.