Downsampling: use a steep low pass filter to prevent aliasing, then reduce the sampling rate by leaving out samples.
Upsampling: insert zeros, then apply a low pass filter to suppress all the new (probably unwanted) side bands. This works better than interpolation because interpolation basically represents a pretty lame low pass filter.
N.B.: filtering can best be done in the digital domain as this is probably cheaper than creating steep analog filters.
The above was inspired by the book & courses "Digitale signaalbewerking"
by A.W.M. van den Enden N.A.M. Verhoeckx