There is a wide range of filter circuits, each with its own set of advantages and disadvantages. All filters introduce phase shift, and (almost all) filters change the frequency response. There is one class of filter called "all-pass" that does not affect the response, only phase. While at first look this might be thought rather pointless, like all circuits that have been developed over the years it often comes in very handy.
Filters also affect the transient response of the signal passing through, and extreme filters (high order types or filters with a high Q) can even cause ringing (a damped oscillation) at the filter's cutoff frequency. In some cases, this doesn't represent a problem if the ringing is outside the audio band, but can be an issue for filters used in crossover networks (for example).
If you are not already familiar with the concept of filters, it might be better to read the article Designing With Opamps - Part 2, as this gives a bit more background information but a lot less detail than shown here. There is some duplication - the original article was written some time ago, and it was considered worthwhile to include some of the basic info in both articles.
Filters are used at the frequencies where they are needed, so the filters described here need to be recalculated. I have normalised the frequency setting components to 10k for resistors, and 10nF for capacitors. This provides a -3dB frequency of 1.59kHz in most cases. Increasing capacitance or resistance reduces the cutoff frequency and vice versa.
Capacitors used in filter circuits should be polyester, Mylar, polypropylene, polystyrene or similar. NP0 (aka C0G) ceramics can be used for low values. Choose the capacitor dielectric depending on the expected use for the filter. Never use multilayer ceramic caps for filters, because they will introduce distortion and are usually highly voltage and temperature dependent. Likewise, if at all possible avoid electrolytic capacitors - including bipolar and especially tantalum types.