Which is faster, copying everything at once or one thing at a time?

If they are all coming from different physical disks, but being written to a solid state drive (flash, SSD) then you can copy them simultaneously.

The physical head movement (latency) is what will slow a transfer from a spinning platter disk. If you were copying from a single HDD, then you would start a single big transfer rather than lots of small ones. The head going back and forth between two or more files that are being copied is what slows the transfer. Of course you would get the same effect if the HDD were heavily fragmented even if you were copying a single file.

If it was between solid state drives, then it would not matter.

If you don't want to wait around for each transfer to finish then use a copy queuing app like Teracopy.

Of course the flash drive write speed would be the bottleneck here, so it probably doesn't matter which way you do it. :)

If destination disk is an SSD, just keep in mind that SSD's are, for what's in concern here, damn fast. (*)

If destination is anything slower (from a flash with not-so-high-write-rate to a SATA3 HDD to a magnetic tape) you should queue the transfers, one at a time.

Your bottleneck will be on destination disk for sure

In HDDs this advised nowadays because, even though there is more than one head in a cylinder, they are not allowed to read nor write concurrently (it has shown no performance impact in real world scenario due to the fact that we can encounter random data in one of the heads that would not allow us to R/W in that head for most of the time being, and if the heads are not completely centered that could lead to disk corruption).

Common sense (and mathematics too!) say that if we have a very very source but we can make the transfer of files in parallel the head seek could (in not many scenarios though) be faster than serialization of the job.

(*) That is, as nnewton states, because there is no head movement. But should we not use the term latency: latency is head movement + disk rotation delay + read time, and in SSD that is simply random read time and does not depend on the previous read position as disk drives do

Using the built-in file copy utility in Windows they are both horribly slow. You may want to consider using an alternative application like TeraCopy.