I have to convert a bmp into another standard format, without losing quality. To which format shall I convert it to?
You already got it in the tags. PNG is another lossless format that is widely supported. Depending on the software you use to create it, PNG images are compressed by filtering and using the DEFLATE algorithm, which means that their file size will probably be smaller than the original BMP.
You can also try to convert it to a TIFF image, but that format offers more features that you might not even need.