So, I know next to nothing about the best efficiency things you can do so I thought I would bother asking: when mining, is it worth it to throw away copper and iron and other low-level metals in favor of higher value ones? Like is it worth it to have an entire cart of silver in favor of throwing out all your iron and copper

Dropping consistently is a lot of upkeep, because copper only stops appearing very late into the mines, and iron even later (which... should not be the case, but that's another can of worms). Generally speaking, rarer materials do give you more xp when you smelt, but it's only worth it if you remember to check your cart regularly and have the time/energy to be nitpicky, or want to save wheel durability and stamina and not make as many trips. (Side note: once you get the enchanted pickaxe equipped on your miner, trips won't consume stamina anymore, so that's always worth working towards! It's a bit of a process, though.)

And then there is the complete flip side where breaking picks and wheels progress two achievements, so people trying to grind those will always be looking for copper equipment because it breaks the fastest. You could probably sell quite a bit of your copper and iron (or better yet, craft the picks/wheels yourself to either grind your own achievements or sell the processed materials), especially in the beginning, and then start dropping it later when you're tired of running back and forth all the time.

Honestly, I almost exclusively mine in the furthest two mines, so I drop all of my copper, iron, and silver. The other ores and gems are worth a lot more XP wise, and crafting with them rewards more XP and can yield profitable items you can then flip on the market or sell directly to the vendor.