Fixing and washing isn't really an issue as long as long as you don't rush it (and your fixer is in good shape).
Newer B&W films such as T-MAX may also have a purple colouring of the film base if the fix is too short. Typically, these films need a longer fix than traditional films, there are a lot of online tips on this.
Running the film under the tap is not the most effective way to wash it. It will be better to use a wash hose for the the tank, or to use the Ilford method of reapeated pouring and agitating. It's also a good idea to let the film soak in some clean water at the end of the wash cycle, for up to 20 minutes. (Sometimes you will actually see darker deposits in the water from the film.)
Wash aids such as hypo clear and flo products can also be useful, especially if your negatives don't come out as clear and clean as you want them to (provided that you have fixed and washed properly).