a few tips from a pilot underway to do the same (Single engine, different aircraft: C152, Beech 36, Cap 10, Piper arrow, Diamond DA20):
1 - Take your time. This is gonna take a while
2 - Start from several places. The weather will trap you for weeks at a time. I’ve started out from 3 places, looking to connect them. I got stuck in the Aleutians for the entire winter due to icing, wind, clouds.
3 - C172 does not have a very good range. You will run into trouble across some stretches.
4 - Use Neofly to help you give a sense of purpose, routing options and fun.
5- Have fun! Take some screenshots
