Assuming all rasters have the same x,y dimensions (number of pixels), that will depend on:
How many bands? More bands = more storage space, so multispectral rasters will probably take up more space
What is the raster data type? Byte data (8 bits, 1 byte per pixel) uses less storage than integer(12 or 16 bit, 2 bytes per pixel), which is in turn smaller than floating point (32 bits, 4 bytes per pixel).
If we imagine a modern multispectral image with 6 bands coded as 16 bit integer values, we have 6 bands * 4 bytes = 24 bytes per pixel. If we then classify this raster and represent class values as simple 8-bit values (limited to 0-255, but who has 256 classes?), we would have 1 band * 1 byte = 1 byte per pixel. So the multispectral raster would be 24 times larger on disk than the classified raster, for the same number of pixels.
On the other hand, if I had a 3-band multispectral file coded in 8-bit, then used these bands to calculate a vegetation index, and stored the results as decimal (float) values, I would go from 3 layers x 1 byte = 3 bytes per pixel to 1 layer x 4 bytes = 4 bytes per pixel, and my file size would increase, even though I have less bands.
The comparison of raster vs vector file size is much more complex. While raster file size is always number of pixels * number of bands * number of bytes per pixel, vector file size will depend on the number of polygons, the amount of nodes necessary for defining each polygon, the amount of attributes stored for each polygon, and the data type of each attribute (byte, integer, float, character). So there is no easy way to predict it.