No, compressed audio requires either a specialized chip (which is expensive) or a very powerful chip. The Arduino microcontroller can't uncompress MP3 on the fly and to keep the shield inexpensive, no mp3 decoder chip is included.
It can play uncompressed Wave files (.wav format). This is a standard format and pretty much every audio program can convert your music or audio into wave format. Make sure the sample rate is mono, 22KHz (or less) and 16-bit (or less). The user manual has instructions on how to convert and adjust files for optimum playback
The best way to determine if the quality is good enough for your project is use Audacity and go thru the steps in the User Manual for converting MP3s (and other files) to 22KHz/16-bit format.
There is no hook-up for a microphone, so there is no easy way to record audio. There is also not enough program space on the current Arduino chips (atmega168) to support recording audio and saving it to the SD card as well as playback.