Performance, ROM, RAM and the number of peripherals is better on the Cortex M3, no contest. Advantages of the ATmega might be 5V operation (plenty of sensors and displays are still designed for 5V), current source/sink capability, complexity and availability in larger pin pitches (DIP/TQFP). The 32 refers to 32kB of ROM, not to 32-bits. An 8-bit MCU might be easier to start with.
The AVR32 is pointless in my opinion, I wouldn't start with it if you don't have anything invested in it. The Cortex M3 is widely available from multiple manufacturers.