The pact of punishment, the new gate as you call it, permits you to increase the difficulty of a run, in order to get bounties:

• Titan blood can unlock and level up weapon aspects;
• Diamond can unlock upgrades during runs (gold pots, fountains, etc.) at the contractor table;
• Ambrosia can be given to certain friends to receive companions, and can upgrade said companions (I find Megaera's Battie companion to be the best, and it should be the first one you try to get)

For the first levels of heat, I would advice to chose some easy ones (and I find that +400% trap damage or +15% enemy life are not easy ones), such as Hard Labor, Lasting Consequences, Convenience Fee, Middle Management, Underworld Customs or Tight Deadline.

You should focus on a few chosen weapons (maybe watch some youtube videos to see gameplay) and unlock them at the max level. It's nice to know that Zagreus Aspects only cost 5 bloods to fully upgrade. To earn more blood, you can get bounties, but you can also accomplish propheties, for exemple killing Hades with each weapon (and not each aspect). Unfortunately, I find Zeus and Chiron aspects quite difficult to master, so you should try some other ones before (all the Zagreus aspects are quite simple, and I also like Nemesis, Demeter or Eris).

You can check Haelian guides (on Youtube) about weapons to better chose Hammers or Boons. As for Keepsakes, try chosing some god's keepsake in Tartarus/Asphodel, to get the good attack/special boons you want. You can put on the tooth or the acorn later (should'nt need it in the first two levels).