On a 60GB drive how many partitions + how big and what file system should they be to install slackware?

It's up to you - all kinds of threads on this. I like splitting off /home, /usr/local (my stuff) and /tmp, /var (variable stuff) and leaving the rest on root. Such a root might be 2-4 gigs depending on how much you put on. Might want to leave a little more for breathing room. /var doesn't need to be very big - half gig is probably plenty, but this depends on what you put there - if you're spooling mail for hundreds of users it's different than if you've got ~/.mozilla for one. /tmp I actually like to make way bigger than it usually needs to be because a lot of apps dump stuff in there that can be huge and it's easier than pointing them elsewhere - gig, gig and a half that's almost always wasted but real useful when I need it. I take as much as I can for /home because that's the point of it all. But it's too variable to give a definite answer. You *can* just make a swap partition and stick everything else on one partition if you just don't want to bother with it.

As far as fs, I like reiser but ext3 is a good solid choice. Again, it's up to you.

Two partitions with one for "/" and one for "home" to have your private data safe in case you screw up the system is usually a good idea - plus a little bit of swap for good measure. I'd do something like:

hda1 about 500-900mb swap
hda2 around 5gig root
hda3 the rest /home

For the filesystem I tried ext2 but was rather annoyed when it had to check and fix broken stuff all the time, especially when I was just beginning with the distro and screwed up a lot at first.
Reiserfs seems more stable and a couple of pros I asked told me the software-companies they work for all have reiserfs on their systems.

I went for ext3 because the 2.6.11 kernel didn't seem to like my reiserfs...
I like to keep it all on the same partition. I think the rule of thumb is that you should make your swap roughly the same size as your RAM.

Basic rule of thumb for swap is double whatever your RAM is unless you have 1G or more then same as RAM is ok because you'll likely rarely need to swap anyway.