Nursery is the fly point Paniola Ranch on 2nd island. You leave 2 pokemon there; if they are in the same egg group and opposite genders they will have a chance of generating an egg every few hundred steps (Nursery girl out front will cross her arms). The female in the pair will determine the species. Ditto can be used as a substitute parent and the child will always be the non-Ditto and this is the only way most genderless pokemon breed. Egg groups are usually pretty logical. Mammals breed with other mammals, dragons and lizards breed together, etc. Serebii and various other sites can help on that front though I'd say 90% of the time the Ditto option is preferred for convenience.
Every pokemon generated from the wild, gifted by NPCs, hatches, etc rolls 6 numbers 0-31 for each of its 6 stats on generation: HP Atk Def SpAtk SpDef Speed. This is the IV, you can consider it like genetics. In the postgame, at the Battle Tree there will be a NPC who, if you have hatched like 20+ eggs, will add an "IV Judge" feature to your PC which will give you an idea of something's IVs ("best" on a stat means it's maxed at 31). Usually when people refer to 4IV, 5IV, 6IV they're citing how many of their IVs are maxed.
When breeding, 3 IVs are chosen at random from the parents' total 12 IVs and passed down directly to the child. With Destiny Knot you pass 5 instead. The same stat can be chosen on both parents, in which case mom takes priority. Also when breeding, pokeball will pass down from the female of the pair (or non-Ditto if breeding with Ditto, or 50/50 between both parents if they are the same species). Also when breeding, if either parent knows a move that falls under the "Egg moves" category for that pokemon, it will be passed down. This is different for each mon. Also again, ability will have a high chance of being inherited from the mother, and Hidden Ability (rare ability found in SOS battles sometimes for most species) can be passed down in a Ditto-parent or same species parent-parent pairing, though in the latter I think the mother needs to have it for the chance of inheriting to exist.
-Parents holding "Everstone" will pass down their nature 100% to the child.
-Parents holding "Destiny Knot" will increase the number of inherited IVs from 3 to 5.
-Parents holding a Power item like "Power Anklet" will 100% pass down the IV of that stat from that parent. This stacks with Destiny Knot but counts towards its 5 inherited IVs; it's not too useful for breeding good IV mons because you lose out on controlling the nature with Everstone but it's worth noting.
All of those items can be bought with BP from Battle Royal dome; Power items are not strictly necessary for breeding but if you don't want to do the lazy method for raising EVs with Poke Pelago you might want them for EV training once you have your good mon. Destiny Knot can be randomly gotten from Pickup mons, and I got a couple Everstones just playing the game so.
Now, for shiny odds 2 things increase it. One is the Shiny Charm, earned for (in this game) getting 100% caught completion for Aloladex (ignoring Magearna). The other is what's dubbed the "Masuda method"; if you breed 2 pokemon from different languages, the shiny chance is increased. So the shortcut to having a good catch-all Masuda-mon (because all hatched eggs become your language/OT) is a foreign Ditto with good IVs. Unfortunately Ditto itself cannot be bred so catching a wild one with 6 IVs is nigh impossible; 4IV Ditto is just time-consuming and 5IV takes a little luck.
Slight tangent, but the SOS mechanic is how this generation "chains" for high IV pokemon in the wild. Long story short, the 31st SOS'mon and beyond in a chain is guaranteed 4 maxed IVs. The last 2 will still be rolled randomly between 0-31 so there's always a chance at it being 5IV or 6IV.
I won't get into EVs right now but tldr it's like working out at the gym. Focusing on beating stuff that gives effort points towards the stats you want to focus on.