First I downloaded all the new packages. I updated them in the order specified.
Then I realised I had a problem - I had not downloaded any of the new kernels! I managed to do this.
While downloading, I had a problem with wget. It seemed to be downloading extra files in parallel directories to the one I wanted! When I researched this problem later, I found that I should have used the "--no-parent" option. (Which reminds me of this Dilbert comic (25/4/1994).
I couldn't get to a login prompt when I rebooted, because the Internet connection scripts were failing in a typically idiotic manner. Rather than realizing they couldn't connect, they just kept on trying and blocked everything else from happening. I rebooted and started the kernel in single-user mode, and disabled them all ("chmod a-x rc.inet1" and so on). Now I could log in.
The biggest problem I had after that was getting wireless internet to work. I realized I needed to reinstall the ndiswrapper module for the new kernel, and I got a package from slackbuilds.org.
"lspci -k" showed me that the wrong driver was being used. I also had to blacklist some kernel modules by adding the following lines to /etc/modprobe.d/blacklist.conf:
I had those lines already, but a new blacklist.conf had been installed.
Connecting to a wireless router worked as before, with wpa_supplicant and dhcpcd.