HDD Slow Error

Hi guys.
May I ask you about some hdd failures question, I'll try to describe as clear as possible.

I just bought a new harddrive, Seagate 1TB, ST3100somethingAS .
And I recenty transferring the content from my old hdd to my new hdd.

Suddenly when transferring, the computer become very slow in copying & windows also very slow. (as in very very slow, 1 single file (around 1 Mega) about 30 second or so.

From what I knew, this condition happen mostly when the harddrive got bad sector, or at least the hdd is failure or soemthing.

Then I restart my computer, I'm expecting some failures found in my hdd. But unexpectedly , nothing is wrong. Usually when that condition happened, the files which copied very slow can't be accessed anymore (from what I experienced). But on this case, everything perfect. So I just continue to copy files.

My question is, does this case only a windows failure, or this means a failure on my hdd ? because I store very important datas. I'm not sure whether it is failure hdd or not, if hdd failure I should change my hdd then.

For more details, I actually copied a large amount of datas when it became very slow (around 20 ,000 files, about 110 giga in size). Is it because I copied too much ? As in I have to transfer little by little ?

I tried to research but no thread give some clear clue.
And one more thing, may I know what is the best software to check whether your harddrive is error or not ?
Because I experienced using chkdisk before, not so useful. Oftenly hdd is failure but chkdisk detect nothing wrong.

Thank you very much for your help. Really appreciate it. I don't have many people having computers in my city, hence it's pretty difficult to find hints..