I build my own banana-to-dupont cables, it's far more convenient and reliable than alligator clips to dupont jumper wires. I've done that in the past, and it's hella ghetto, you end up with stuff coming loose, or shorting out… it's OK in a pinch but you'll be happier with properly made cables. Making test leads is a great way to practice your cable termination/splicing/dressing, by the way.
Forget alligator clips, everyone here is telling you "I have 'em and practically never use 'em". Go minigrabber instead.
I couldn't even tell you when I last used spades on test leads.
Forget the no-name ones on amazon. Go with top quality name brand (Fluke/Pomona, Mueller, Probe Master, Cal Test, Stäubli, Hirschmann). There's another thread on here from someone asking about this, because it turns out the cheap banana plugs on his Amazon-sourced no-names are wiggly and caused problems.
BNC is for high-frequency (but low current!) stuff, which is why your oscilloscope will use it (as do signal generators and the like). For now, don't buy any extra BNC stuff, the probes with the scope are more than enough to get started.
Lab power supplies use one of two things, generally: safety banana jacks (accept sheathed or unsheathed banana plugs), or banana jack binding posts (accept unsheathed banana plugs, large spades, and bare wire, but not sheathed banana plugs).
But ultimately don't sweat this too much. Start with the basics (banana-banana, banana-minigrabber, and minigrabber-minigrabber; I suggest 18"-24"), and buy yourself some flexible wire and some quality banana plugs (any of the brands I listed above is great) and make additional cables as you discover the need for them. I highly recommend the Pomona 1825 banana plugs, because they're solderless with a set screw that requires no disassembly, so you can easily and quickly build a custom cable with them for a specific purpose, and later reuse the plugs for some other cable as your needs change. (Many other plugs hide the set screw under a sheath that you have to wrestle off first.)