According to Samuel Eliot Morison, the US Navy lost 2 escort carriers, 1 destroyer and 1 destroyer escort in the battle to naval gunfire. Yamato claimed hits on all but the DE, USS Johnston - although there has been no irrefutable confirmation on who hit what. As far as damage to ships that survived the battle goes, large caliber hits did some of the damage damage, but again, who hit what is conjecture only. There were four battleships in Kurita's force and all participated.
At the beginning of the action (0558) Yamato fired 4 salvos from her forward turrets at an aircraft carrier (Gambier Bay?) at a range of 20 miles and claimed at least one hit. The target was seen to be smoking so she shifted fire to a second carrier, but it was "immediately obscured by smoke". At 0651 a "cruiser" (Johnston?) emerged from the smoke screen and was taken under fire at a range of 10 miles, a hit was claimed and the target was seen to be burning. At 0654 Yamato turned north to avoid torpedoes which took her out of the battle. (This from www.combinedfleet.com . Japanese log times seem to differ from US which say fire was opened at 0648.)
As was said, since the ships sank nobody can say for sure who hit what. Escort carrier Gambier Bay, while dead in the water, was finally sunk by three cruisers firing at point blank range. Destroyers Johnston and Hoel and destroyer escort Samuel B Roberts were sunk by gunfire from battleships, cruisers and destroyers. Escort carrier St Lo (ex-Midway...bad luck to change the name!) was sunk by a kamikaze an hour or so after the close of the surface action.