Will the HDD be faster if I plug it into SATA 3 rather than SATA 2?
Do you know any software which will test the read/write speed as I'll try SATA 2 and 3 port to see which gives me better speed
No it won't be.
don't matter, mechanical hard drives cant even reach full speed of sata 2, even if it states sata 6 gbps on that seagate it will never reach those speeds, so either way is the same, i dont think u should waste ur time testing the 2 ports as they will be the same. for an ssd, it is a different story.