I've done some PIC programming - mostly because I liked the idea the chip were only a dollar or two. However, for a beginner, making a decision solely on price is premature optimization.
Programming in assembler is an experience. You basically have to learn about 100 concepts before you can blink an LED. (Watchdog timer, reset pins, 8-bit counters/overflows, delay loops, hex, binary, bit-masking, interrupts, interrupt service requests, IO ports, etc.) It's all very educational - and a great feeling to get so close to the machine - but being able to code something in C will hide some of this complexity so you can focus on results. For this reason I would say go with the AVR. (And I believe the prices are now closer to PICs.)
Also: If you're interested in getting things done (and don't mind spending ~$30) check out the arduino. A guy selling them at my local electronics shop was saying he's selling tons of them to art students. (It uses the IDE from the Processing project, and compiles code with avr-gcc.)
Update: Fixed comment that Arduino runs interpreted code. Also updated the approx Arduino price.