My Internet's been slow lately and I did a SpyBot update just before my scan (it'd been a while), so I saw the following updates available:

Detection Rules: Dialers
Detection Rules: Trojans
Detection Rules: Malware
Detection Rules: Keyloggers

and a few others, but beside those it said under info once I started downloading "!!! badchecksum !".

Should I do a HJT or is there something else you'd recommend?

"Bad checksum" warnings generally indicate that the downloaded file was corrupted. The file in question would not have been installed, so you would have to redownload that file again.
Regards,
John

You can also select another download location.

Do not click "Search for Updates" again.
Pull-down site selection menu (between the "Search for Updates" and "Download Updates" buttons)
Select another server
Click "Download Updates" again.

If you keep encountering error messages while downloading the updates, you can always manually download them from here