First, backup important data (But you knew that already)
What you are talking about is cloning (or ghosting). In the Windows world, they sell programs (eg Norton Ghost) to do this. Linux has free tools.
The most basic command is dd. This will do amazing things (including totally wiping out your system---did I mention backup??....
)
Suppose you have a set of partitions that occupies a 40GB hard drive. The entire thing can be copied to your new drive, and it will behave exactly as the old one--except it will have a whole bunch of space left over.
If the 60GB is full, then you can do a dumb clone--ie copy *everything* on the old to the new. If it is not full, then you may wind up wasting some space (But--nothing lost compared to your status quo)
Use fdisk to see where the partitions end, and then go bit beyond, to be safe. For example, my 40GB Linux drive shows 4866 cylinders of 8225280 each= 40024+ Mbyte. The disk is supposed to be 40027+, so I might actually copy--eg--2444 blocks of 16384 bytes each--just to give a litle margin (In dd, you can use any block size that is a multiple of 2)
So, for my example, copying the entire contents of my 40GB drive to **anything larger**, the command would be:
dd if=/dev/hda of=/dev/hdb bs=16384 count=2444
(I've assumed hda and hdb--yours may be different)
This approach might give an error message when it tries to read beyond the end of the source drive.
If you first backup, and then save the old drive until the new one is up an running, then there is minimal risk in trying this.