I like your original suggestions for Favored Terrain as is. They all feel flavorful and related to the named terrains. I agree that making them all resistance based feels bland. The only thing I would change is the language for the Coast (to "You gain a swimming speed equal to your walking speed") and the language for the Mountain (to "You gain a climbing speed equal to your walking speed"). Functionally it is the same as what you have, but the language is more consistent with other features that have the same effect.
Good job on these.

Favored Quarry is more problematic. Since Rangers would get 3 Terrain choices and 3 Quarry choices, it doesn't seem like a good idea to give resistances for both features. Also, some of your initial suggestions would allow a 1st level character to cast a 2nd level spell, which I think is a bad idea. On the other hand, I agree that some of Kane0's suggestions for this feature don't feel like that much of an upgrade for the Favored Foe abilities in the PHB. My suggestions to try to balance these abilities (combining your ideas and Kane0's ideas where I can):
Aberrations: You have advantage on saving throws to end ongoing effects.
Beasts: You gain proficiency in Stealth and your proficiency bonus is doubled for any ability check that uses your Stealth proficiency.
Celestials: As a bonus action you can imbue your weapon attacks with magic. For the next 1 minute all of your weapon attacks count as magical for the purpose of overcoming damage resistance or damage immunity. Once you use this ability you must complete a short or long rest before you can use it again.
Constructs: When you interact with a creature that is controlled by another creature you can attempt to find a loophole in the creature's orders. Make a Charisma (Deception) check opposed by the creature's Wisdom (Insight) check. If you succeed you convince the creature to that taking an action or refraining from an action is consistent with its orders even if it conflicts with the intention of those orders.
Dragons: Attacking at long range doesn't impose disadvantage on your ranged weapon attack rolls.
Elementals: As a bonus action you can enhance your weapon attacks with elemental energy. Choose acid, cold, fire, or lightning damage. Once per turn for the next 1 minute you can 1d4 of the selected damage type to the damage when you hit with a weapon attack. Once you use this ability you must complete a short or long rest before you can use it again. (NOTE: You should not use the UA spell Arcane Weapon. It did not make the transition from UA to Tasha's Cauldron, and this was almost certainly because when you compare it to the level 3 spell Elemental Weapon it is too powerful for a level 1 spell.)
Fey: You have advantage on saving throws against effects that inflict the Charmed or Frightened conditions. You also have advantage on Wisdom(Perception) checks to detect hidden creatures. (NOTE: Giving proficiency in an additional Saving Throw is too powerful for level 1, and is not balanced against the other Favorite Quarry options.)
Fiends: You learn the spell Protection from Evil and Good, which counts as a Ranger spell for you, but does not count against the number of ranger spells you know. You can cast it without expending a spell slot and without requiring concentration. You must complete a long rest before you can cast it in this way again.
Giants: You can move freely through other creature's spaces, without spending extra movement.
Humanoids: Each time you choose Humanoids (at levels 1, 6, and/or 14) choose one of the following skills: Deception, Intimidation, or Persuasion. When you make an ability check with your chosen skill you can add your Wisdom modifier to the roll. You must choose a different skill each time you choose Humanoids.
Monstrosities: Whenever you cast a spell, you can treat Monstrosities as Beasts or Humanoids for the purposes of determining if they are a valid target for that spell.
Oozes: Unfortunately nothing is coming to mind for this. I'll have to think some more and see if I can come up with anything.
Plants: You have advantage on ability checks to resist or end effects that impose the restrained condition. (NOTE: I was thinking of entangling vines when I came up with this.)
Undead: Whenever an effect would reduce your maximum hit points, your maximum hit points are reduced by half the amount the effect would normally reduce them by. Incorporeal creatures can not move through your space and when you attack an incorporeal creature your weapon attacks count as magical for overcoming that creature's damage resistance or damage immunity. (NOTE: these abilities are very situational, but when they apply they can be very helpful.)