If you want to make induction recordings, you'll need a recorder that has a good enough preamp to boost the signal from the pickup to a usable level for the recorder's electronics.
Digital voice recorders cost $20-60 and are meant for audio notes/dictation, recording lectures & interviews, etc. They work with external condenser mics by supplying a small voltage to a preamp that is built into the condenser mic. Induction pickup don't have that internal preamp so they may or may not produce a strong enough signal to produce a good recording with a digital voice recorder. You may ask "Can't you just turn up the volume if the signal is weak?" Yes, increasing the gain will boost the signal level but it will also boost any inherent noise in the recording from the recorder's electronics as well. What you want is a good high signal-to-noise ratio (SNR) and not just a louder recording.
Digital PCM audio recorders are anywhere from $100-$1000 and are targeted at audio professionals like musicians, recording engineers, etc. The preamps in these types of recorders usually have enough gain to work with induction pickups.
SamH said:
Just something good enough to listen to the song with limited background noises/clearer music.
Unfortunately that is very subjective. Also, "limited noise" and "clearer music" compared to...? What are you using as a basis of comparison?
If you have music format conversion software on a computer, try picking some favorite tunes in various styles of music and converting them to a series of MP3s with progressively lower bitrates. E.g., 320 -> 256 -> 192 -> 128 -> 64 -> 32 kbps. Listen to the converted tracks and try to find out where the decrease in bitrate really does bother you.