Are TI TMS320C6x and TI TMS320C2x compatible?
There is a big difference between C6000 and C2000.

Mainly, C6000 is a DSP (very high speed, starting from 456 MHz) and, as it is a DSP, it's more focused on signal processing. 

C2000 is more like a general MCU (freq. range in 60-200 MHz). PWM and ADC is the main core of C2000 (strongest side). C2000 is focused on power electronics (motor control, power inverters). So I would say they are very different, although I personally have experience only with C2000.

Anyway, it's quite a challenge to learn any of them in 2 months, both are complex. But if you need to learn C2000 then go for C2000 and definitely not for a C6000. On C2000 you need pure C for 90-99% of your code.

I would start with a TI C2000 workshops as a starting point. But you definitely need a launchpad (real chip) to try your code. The easiest and simplest MCU to start with is is TMS320F28027 (link to the launchpad).
You can ask specific questions directly on TI C2000 forum (btw, there was a similar thread a couple of days ago).