These are the best enchantments to have on your armour and tools.


Armour
Armour: Protection IV, Unbreaking III and Mending (plus additional enchants for specific items)

Helmet: Aqua Affinity and Respiration III

Boots: Depth Strider III and Feather Falling IV
Optional: Blast Protection IV on one piece of armour to better survive creeper surprise attacks.
Optional: Thorns III on one piece of armour to damage attackers.
Optional: Frost Walker II on boots (instead of Depth Strider)
Frost Walker will let you walk on water, or walk on Magma Blocks without damage.
Personally, I find Depth Strider to be much more useful.

Pickaxes
Pickaxe #1: Silk Touch, Efficiency V, Unbreaking III and Mending
Pickaxe #2: Fortune III, Efficiency V, Unbreaking III and Mending

Swords
Sword #1: Sharpness V, Sweeping Edge III, Looting III, Unbreaking III and Mending
Sword #2: Smite V, Sweeping Edge III, Looting III, Unbreaking III and Mending
I also like Knockback I (Knockback II is too much).
I personally do not like or recommend Fire Aspect on a sword.
Since swords are a close combat weapon you are likely to end up on fire yourself.
If you want to set mobs on fire, do it with a bow. And speaking of bows...

Bow
Bow: Power V, Unbreaking III, Punch I, Flame, Infinity and Mending (see note below).
Both of my current 1.12 worlds were created in an earlier version, so both of my bows have Infinity & Mending.
For bows created in 1.12, you will have to choose one or the other (you can no longer have both enchants in 1.12).
Personally, I'd probably go with Mending, since arrows are easy to craft, and can easily be acquired in large quantities with a multi-mob or skele grinder. On the other hand, an AFK fishing farm will give you plenty of enchanted bows, so it will be easy to craft another OP bow if you go with Infinity and wear out your old one ...

Axe
Axe: Sharpness V, Efficiency V, Unbreaking III and Mending
Optional: Fortune III or Silk Touch
Silk Touch is great for mining bookshelves and melons.

Shovel
Shovel: Efficiency V, Unbreaking III and Mending
Optional: Fortune III or Silk Touch
Fortune will give you more flint from gravel (100% with F3). Silk touch will give you none.
Silk Touch for mining grass, mycelium, podzol, etc...

Elytra
Elytra: Unbreaking III and Mending
These enchants are pretty much a must on Elytra (their default durability is quite poor).

Fishing Rod
Fishing Rod: Lure III, Luck of the Sea III, Unbreaking III and Mending
Some may suggest that having less luck or lure makes it better for certain things.
I have experimented with that and it is not true.
A maxed out rod is better, period! End of story. Just do it!

Optional
Optional: Shield with Unbreaking III and Mending
Shields are so inexpensive to make that the value of enchanting them is questionable.
That being said, mine is enchanted...