Mii Fighters can be created directly in Smash Ultimate, which is handy since you can't create them on the Switch menu. Mii outfits are still collectable in Switch.

All custom moves have been removed with the exception of Mii Fighters which have the same custom moves as in Smash 4. Their custom moves are edited in the settings used to create Mii Fighters.

Spirits can be used in local multiplayer matches and the World of Light. They can add a larger variety of buffs, stats modifications, and even nerfs. They function somewhat similarly to stickers in Brawl but with a larger range of effects and they can be removed from a character without drawback (removing a sticker causes you to lose the sticker).

In addition to spirits is Dojo training. There are various Dojos obtained through playing the World of Light which each providing a range of stat modifications that can be applied to spirits, for example increased jump and weight but decreased speed and attack. One Dojo (Slowpoke) can remove other Dojo stat modifications.

Amiibos function similarly to Smash 4. Any amiibo for any of the playable characters, including those for Smash 4 and those not created for Smash, work in Smash Ultimate: a list of these amiibos is here. Certain amibos also grant you a spirit that corresponds to the amiibo, also contained in that list. Contrary to the rumours, amiibos don't unlock characters instantly.

Gym/Training

For starters, there is a node on the skill tree at the far bottom left called Activities ↑. It "increases Gym and Dojo efficiency, and increases loot in Explore." So it is a no-brainer to go for this node first to fully impact all your expeditions and gym outputs onward.

For gym training, it is definitely important to slot fighters that have the red upwards icon to the right of their cards. As you can see here, the Bandana Waddle Dee has the bonus icon next to its name, and from experience it definitely levels faster than spirits with the negative icon:

Expeditions

As for expeditions, it seems from personal testing that there are only two things that factor into loot: the rank of the spirit (# of stars), and whether or not said spirit has positive bonuses to the expedition. Here's two screenshots of two expeditions I made with Toadette's expedition. The first consisted of all 1-star rank spirits:

And the second consisted of all Legend (4-star rank) spirits:

Extensive testing showed that level factors seemingly little into the final result. As you can see the difference between the 1-star and 4-star expedition was nearly DOUBLE.

So to sum it up?

[4-Star Rank + Positive Boost] > [4-Star Rank] > [Positive Boost]