First of all: shinies have a visual and sound effect when entering battle, so pretty hard to miss. And the name will be, for example, "Shiny Eevee". If you rename it, it'll just have the name you give it, without the "Shiny" part.

Now to hunting.
The base chance is 4x less. So around 1/30k.
But you can use the Sweet Scent move to call hordes, where you encounter up to 5x mons at a time, making your chance to see at least one shiny per call around 1/6k.
This consumes 5 PP per use, so if you PP Max it you have 6 uses till you need to recharge — either at PC or using leppa berries.
Some people try to stay close to a pc when douing this if they want to recharge cheaper.
Having multiple mons with Sweet Scent is useless, because PPs are deducted from all of them at once.
Leppas are cheap, when you start shunting seriously you should be able to afford them.
Donator ticket adds 10% shiny chance.
You can buy it for ingame pokeyen on GTL.
Shiny charm is an event item here. Between events you can only buy it from other players. +10% chance to the one that uses the charm and +5% to those who are linked with the user, for 1 hour.
Consumable.
This covers basic hunting for horde shinies.

If you want a rare shiny that doesn't have a good horde or even a decent single encounter, you probably need to breed.
If you decide to go this way, you need to know that unless the mother is your OT, the shiny will show "Unknown OT".
If breeding with a Ditto, the non-Ditto mon is always the mother (even if male).
If breeding two non-Ditto genderless (you can breed these here if they are in the same evo family), you need one of them to be OT.
Here you lose both parents when breeding, the system is different.
https://forums.pokemmo.eu/index.php?/topic/49440-the-breeding-guide/
But it's not really relevant if you're just breeding for a shiny, not caring for its stats.
Breeding for shinies is why shіtto (shіt Ditto) market exists.
You catch a bunch of normal mons that you want a shiny of, for OT.
Then you breed them with a box of shіttos, while charm and donor ticket are active, trying to put out as many eggs as you can before the charm runs out, as it's just an hour.
And then you take your magma armor or flame body egg hatcher and start hatching.
This is basically everything you need to know about getting shinies.

Wild caught shinies are guaranteed to have at least 1x31.
Only giftmons are shiny locked.
You can use other shinies to improve the stats of the ones you like.
Rules for stats when shiny breeding:
https://forums.pokemmo.eu/index.php?/topic/108239-how-does-shiny-breeding-work/&tab=comments#comment-1661994