Hi Andrew,
I've flown almost exactly the same route over the Xmas period except I flew to Luton with Easyjet rather than LGW.
With regards to transferring, just go to baggage reclaim, collect your luggage, then follow the signs to departures and check in with the domestic airline (I assume Pegasus). Since you are transiting with 2 different airlines, you have to collect and re-check in. It's no hassle though as the whole process is under cover.
Finally, Sabiha Gokcen is one of the nicest airports I've been to. It's all brand new and not too busy. Certainly, compared to Ataturk, there is no comparison.
Have a safe trip. Any problems, PM me.
Regards, Champ
P.S I'm living in Izmir