If I have an astral heron, and my second card is a svellsongur, will it also pay for the equip power cost when I play it and equip it to a unit?

No. Astral Heron's discount applies to Svellsongur's play cost only — the Equip cost is a separate activated ability cost you still pay in full.

When you play Svellsongur (3E + 1 Calm), Heron's [2][A][A] reduces it to effectively 1 Energy (the [A][A] covers the Calm Power). But Equip ([1][C]) is an activated ability keyword (Rule 818.1), not an additional cost of playing the card — its cost is paid when you activate Equip afterward and is untouched by Heron's "next card costs less" effect (Rule 356.4). Rule 356.4 only discounts costs summed while playing the card (base + additional costs like Accelerate/Repeat); activated ability costs aren't part of that total. (Compare Weaponmaster, which explicitly discounts Equip costs.)