METAFONT produces from its run three files, a metrics (TFM) file, a generic font (GF) file, and a log file; all of these files have the same base name as does the input (e.g., if the input file was cmr10.mf, the outputs will be cmr10.tfm, cmr10.nnngf and cmr10.log).
For TeX to use the font, you need a TFM file, so you need to keep that. However, you are likely to generate the same font at more than one magnification, and each time you do so you'll (incidentally) generate another TFM file; these files are all the same, so you only need to keep one of them.
To preview or to produce printed output, the DVI processor will need a font raster file; this is what the GF file provides. However, while there used (once upon a time) to be DVI processors that could use GF files, modern processors use packed raster (PK) files. Therefore, you need to generate a PK file from the GF file; the program gftopk does this for you, and once you've done that you may throw the GF file away.
The log file should never need to be used, unless there was some sort of problem in the METAFONT run, and need not be ordinarily kept.