I know sudo lshw is used to list hardware. What would I look for in the output of this command to tell whether I can use a SATA III hard disk (versus whether I'll only achieve SATA II or SATA I speeds)?
Is it under *-storage, or something higher up, in the motherboard or something?
lshw or lspci do not directly tell you which version of sata is supported. You need to check the shown chipset against its specification, which you'll have to google. As an example I use
lspci | grep -i sata
to find out which sata controller I have. The output is in my case
23:00:1f.2 SATA controller: Intel Corporation 82801IBM/IEM (ICH9M/ICH9M-E) 4 port SATA Controller [AHCI mode] (rev 03)
This means I have either the ICH9M or the ICH9M-E controller. Googling for the former yielded a wikipedia article: http://en.wikipedia.org/wiki/List_of_Intel_chipsets telling me that it supports SATA 3.0 Gbit/s, hence SATA revision 2.0. hth :)
lshwyou see all your hardware. With that information you can look up your mainboard model and with that you can look up the SATA speed on the internet under hardware specifications of that product.