I use Heedless Resurrection on my Kog'Maw, Caustic. Do I first resolve the spell by playing a unit from trash, or do I first resolve Kog'Maw's Deathknell?

Kog'Maw's Deathknell resolves first, then Heedless Resurrection plays the unit from trash.

Here's the sequence:

1. You announce Heedless Resurrection and move it to the Chain as pending.
2. During finalization, you kill Kog'Maw to pay the additional cost. Kog'Maw's Deathknell trigger is added to the Chain on top of Heedless Resurrection (per Rule 428.1.a.1.b).
3. Both items finalize — Heedless Resurrection first (it was appended earlier), then the Deathknell trigger.
4. The Chain then resolves LIFO: Deathknell resolves first, dealing 4 damage to all units at that battlefield.
5. Heedless Resurrection resolves second, playing the chosen unit from your trash.

This means the resurrected unit enters play after the Deathknell damage has already happened, so it won't take that damage.