There are two sides to this, depending on whether you plan your path or not:
Because the Clockworks is constantly moving, even when you're progressing through a level, you have the option of deliberately waiting to step onto the elevator in order to reach desired levels. Each of the levels has a (usually) pretty obviously descriptive name reflecting the types of monsters you will encounter, and since gear with fewer resistances is stronger against each source, you can "stack" your resistance to a select few and plan out your path for maximum resistance. They hide the option to show the gate map under the upper-left menu with no way to hotkey it, but it will show you where the elevator is currently pointing.
For the more casual player who expects to face damage from a variety of sources, pieces with multiple resistances will help negate more damage on average than a specific piece. Because they have more individual resistances, there is a greater probability of entering a level they can resist some of the damage from, rather than limiting all of your potential resistance to a single source and missing it.
The bottom line is that very few sections of the game actually depend on your gear's type enough that it makes a significant difference. As long as you have enough stars on your gear and a fair amount of experience with the enemies' tactics, you can just pay attention to your "threat meter" to the lower-right of the minimap.
Well, the overarching goal of the game is to get down as close to the core as possible. To do this, you need to upgrade your gear, which you do by going through levels, getting crowns and such, which you can use to buy recipes and CE for crafting (You'll also need certain materials, but those are a minor problem by comparison.)
As far as the plot goes... well, I'm afraid that's not a major point of the game. There's certain boss levels you can go through (The Snarbolax in tier 1, the Jelly King and Roarmulous twins in tier 2, and the Firestorm Citadel in tier 3), which have recon modules at the end which give you a bit of info regarding the setting and Alpha Squadron, the recon team that went to the core and disappeared. The other bit of plot is down at the core (the end of tier 3), there's a team of NPC knights there doing research, and that's about it. Hopefully we'll get an update soon with some info about what the core actually contains, but who knows?
As far as the mineral gates thing... the type of minerals you dump into the gate will affect what kind of levels the gate contains when it comes out a day or so later. It's nothing that really affects the plot in any huge way, and it takes a ridiculous amount of minerals to even change the theme of one group of levels.
However, keep going, the gameplay becomes a lot more interesting once you get down into tier 2 and the enemies start posing an actual threat, as then you need to start using some actual tactics instead of just repeatedly pressing your mouse button.
Best Answer
As you may or may not know, the gates in spiral knights cycle every 8 days, moving right to left. Every other day the leftmost gate vanishes, the leftmost mineral deposit gate becomes locked and becomes the rightmost gate, and a new gate spawns on the far right.
Every time a new gate enters "circulation", your Knight is "unattuned", and must pay a crown tax to jump to tiers 2 or 3.
The tax is removed when you take the elevator down from floor 7 or 17 to Moorcraft Manor or Emberlight respectively, and will stay removed for the lifetime of that gate.