Hey there. Season 12: Complete the Hunter and the Hunted mission on the Moon. You may have to play through the first campaign mission of Forsaken to unlock the Tangled Shore. Season 13: Go to Nessus and run the Battleground to obtain the Bell of Conquests (you may have to do this on an alternate character). Visit the H.E.L.M. and interact with the War Table and you should receive the Challenger's Proving quest.
i am going from memory here, but there is a mission on the moon to kick off season 12 and there is a battleground mission on nessus to kick off season 13.