For this I can recommend my trusted TASCAM DR-07 Digital Recorder.
It can record either mp3 or wav Format on SD Card and has a
USB Interface to connect to a Computer to Backup your Recordings.
It operates on Batteries or with an optional (you have to buy it
separately

) Power Supply.
It's small enough to be fit in a Gigbag, is easy to use and has some
usefull Features like "Variable Speed Funktion", Limiter, 40/80/120 Hz
Filter, Pitch Control -50% to +16% etc.
The Sound with the Built-In Mics is OK when I put the DR-07 on a Guitar Footstool appr. 10cm in Front of the Speaker of my Cube XL80.
I bought the DR-07 may be a Year or two ago for ~140 Euro.
I can't remember what I paid for the Power Supply.