Normally I do this kind of transfers by rsync'ing between the two hard drives, so your hunch about either rsync or cp -avx looks more than right to me.
In order to boot from the new disk as you did from the old one you just need to reinstall grub in the new boot sector (I'm assuming you're using grub here).
For RedHat based Linux distros (CentOS, RHEL, Fedora) you can do this by executing as root
grub-install /dev/hdX, just telling where your new hard drive is.
For Debian systems you'll need to execute
Reinstalling grub sometimes is a bit of trial/error so be careful.
If you want to do it the manual way you can do so by following this steps:
Check that your new disk appears in grub
Once you know that the new drive is in device.map from the OS booted from the old drive execute the following commands
grub> root (hd0,0)
Filesystem type is ext2fs, partition type 0xfd
grub> setup (hd0)
Checking if "/boot/grub/stage1" exists... no
Checking if "/grub/stage1" exists... yes
Checking if "/grub/stage2" exists... yes
Checking if "/grub/e2fs_stage1_5" exists... yes
Running "embed /grub/e2fs_stage1_5 (hd0)"... 15 sectors are embedded.
Running "install /grub/stage1 (hd0) (hd0)1+15 p (hd0,0)/grub/stage2 /grub/grub.conf"... succeeded
BE AWARE: You need to substitute (hd0,0) with the appropiate disk and partition for your kernel images
If your kernel is in partition /dev/hdX1 -> (hdX,0)
If your kernel is in partition /dev/hdX2 -> (hdX,1)
And so forth...
Also substitute hdX with the drive as declared in your device.map, so if you have your drive declared as (hd1) you need to make it (hd1,X)