there's actually two type of "animals"
this is right in the civpedia under the units tab or unit category tab (inside the unit category tab should have it right there for u)
unit category tab in the civpedia:
animals
*beasts* <-see
melee
mounted
archery
naval
recon
seige
arcane
discipline
1. animals. they can NOT enter your culture/territory. unless they are owned by another civ. (beware some mean-sneaky humans players who use subdue animals from scouts/hunters and wipe u out all your units with their invisible spiders....V.V)
2. beasts. (they are like super animals). i'm not sure if these can enter your culture/territory or not. (i wanna say that NONE of them can, unless again owned by another civ. but i'm not certain)
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hills giants are either animals or beasts. i wanna lean toward them being beasts (makes more sense than animal...but i can't remember what they actually are and too lazy to look up). but whether they are beast or animal, they can NOT enter your culture/territory, unless they are owned by another civ. as has already been mentioned.
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the recon units' (scouts, hunters, and the ranger path, but NOT the assassin path) gets a bonus against BOTH (i think. this could be wrong or changed) beast and animal (most just know of course their bonus against animal). however, the scout is too weak (2 str) to kill hill giants despite the bonus against them. hunters have a hard time too (unless they got some exp/lvl/promotions), but ranger+ waste hill giants.
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grigori dragon slayers are NOT the only unit with a bonus against dragons, the recon and ranger line units get a bonus against dragons too since dragons are beasts.
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easiest way to kill hill giants is to simply wait (get higher military techs and units) until u got a strong enough unit (str wise. or slightly less str but enough promotions to make up for it). an axemen by it self will usually lose against a hill giant. however, with some promotions or copper or (enchanted blade from an enchant lvl 1 spell/promotion adapt), the axeman can win. or simple use 2 or 3 axemen or hunters or horsemen. etc...i think u get the idea...
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btw... it seems like u might be trying to explore around with units....don't it's not really worth it. early game the units (animals=wolf, lion, tiger, spiders, bears, etc and beasts=hill giant?) outside your culture/territory are much more powerful than the units u got at the moment. it's not really profitable trying to explore for the most part. it's much better to be using your city production for things that don't go to waste from some spider or bear or hill giant or gang attack of lions/wolves
however, if u are subduing animals than this is profitable and it is the tactic/strategy u are intentionally doing/using.
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(i actually had this exact same problem when i first played and took me forever to figure out how to build them....finally i learned u use great people to build them..but then had another road block...didn't know different holy shrines required different great people lol. so i'm very sympathetic/compassionate to this question u have)
what do u mean by shrine? the holy shrine (the religions "palace-like" building) or do u jsut mean its basic temple building ?
the holy shrine is called the necronicon
i can't at the moment remember the name of its basic temple building though (sorry). but u jsut build it once u can. nothing tricky like the holy shrine.
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as for mana types/spells, as u said it is subjective based on situation and personal preference.
well first off, ask yourself, WHAT DO U NEED/WANT ? than search the civpedia for the spell that does it
(it's actually easier if u search by the promotion tab and look at the mana types within it in the civpedia, instead of the spells and abilities tab in the civpedia)
basically u wanna look mainly at the lvl 1 (though u can look at lvl 2 also but that requires mages and is a bit farther off. lvl 3 is too far off, unless u spefically need it for your strategy or u wanna try them out) spells that the adapt can use and simply build the mana node of those spells that u want/need the most.
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one final suggestion,
for the most part it's better to NOT stack up on a or only a few mana types (for example: getting 3 air manas)
it's much better to try to get as many mana types as u can (for example: getting 1 air mana, 1 water mana, 1 fire mana) since this gives u more spells and variaty of abilities.
only in competitive games, and when u know what your doing with a already well known strategy/tactic for your war-military force and magic u want to use with it, THAN do u want to stack up on a or a few mana types (for example: if u plan on using fire affinity units, than the more fire mana u have the more powerful those units get)
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finally a great way to learn everything about FFH2: (make a custom game)
easiest difficulty level, inland sea map, low sea level, whatever map size u want (though i'd stay with the lower sizes, since u do have barbarians option on or u should anyways), have barbs be in the game (so u can practice and learn military-units and military magic-units) but NOT raging barbs, and NO other civs. choose whatever civ/leader u want.
it's kinda boring, but it's a great way to see and try and test and learn everything.
if it's too boring add 1 other civ, raise the difficulty, add more civs, have raging barbarians on, agressive ai, etc...
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last point about what magic, mana types/spells, to choose:
think of the spells in terms of categories:
(it makes it easier to understand and choose/decide what u need/want)
(i wish some one would do this for us in the civpedia. make a spell category tab jsut like there is a unit category tab)
"terraforming" spells: change the land (vitalize=lvl 3 nature spell, blaze=lvl 1 or 2 fire spell, spring= lvl 1 water spell, etc...)
"summoning" spells: makes summon combat units (don't have memorized, most lvl 2 spells are/have summon spells)
"unit enhancement" spells: enhances units (haste=lvl 1 body spell, enchanted blade, lvl 1 enchant spell, etc..)
"city" spells: spells that when the magic unit casts the spell and stays in/on the city tile, that city gets bonuses (inspiration=?, don't have these rememorized....)
"direct damage combat" spells. (fireball..etc...)
etc..hopefully u get the idea...