I would not go with swap at 3x RAM. More likely 2x.Depending on how old
the system is, the usual recommendationis to do 2x the ram- up to a max setting
of 4 gigabytes at the most.Some of my older Linux buddies say you don't need
even 2x the RAM,but I usually do somewhere near it on any install that I do.
What partitions do you need? Depends on taste really.The preferred popular
partitions you have already named.Sizes.. again, it's preference. I usually
create a root partition ofanywhere between 8 to 12 gigabytes. This again
depends on howmany files/apps you want to store in partitions other than /home
For /home - I would give it as much drive space as you can.One approach can be
to do the partitioning in this order.1) set the swap space - (again, 2x the RAM
or a little less, should be ok).2) set /giving it maybe 10 gig or so.3) give the
remaining drive space to a partition for /home
Some, by the way, create a /boot partition - if you want to do this (kernels and
grub get installed there) you can, but it really isn't necessary. If you do
this,you can set the size small (500 meg at max should be fine)
Some people also create /optbut again, for a basic Linux system this isn't
really mandatory.
The list goes on. But I'd stick with the basic 3 you already mentioned (/,
/home, and swap).

Why not use automatic install and let Ubuntu do it's thing ?

First, the swap: two times memory is good. If you have a lot of memory (4 GB or
more), just over one time memory is fine.

Second, Root: for most people 20 GB is more than enough. If you really like
playing with new software, you might want more. If you are setting up a simple,
stable system 6 GB is more than enough. You want "more than you need" because
it's a disaster if it runs out.

Home gets the rest. I have a lot of videos, so I have a large hard drive. If
you just do email and a bit of word processing, a few GB is enough. (I can back
up all my non-media files on one DVD, and I have files from 15 years ago.)

I agree with most of what is said except be careful with too much swap. More
is less. It can actually slow down your system because disk caching is
slower than RAM.

I tend toward not exceeding the size of your RAM. Certainly don't go beyond
2 or 3 GBs no matter how much RAM you have.