I have installed Ubuntu by manually specifying the partitions /, /swap & /home. I am new to Ubuntu so I would like to verify whether these partitions are working properly.
df command shows how much free/used space is on each mounted filesystem, and the
mount command shows what filesystems are mounted. You can use either to make sure /home is mounted.
free shows how much used/free ram/swap you have, so you can use that to verify that your swap partition is mounted, or
cat /proc/swaps for a simple list of mounted swap partitions.
A gui based tool is also an option.
GParted provides a graphical view of devices and partitions. It might already be installed on your system:
Applications | System Tools | Administration | GParted
If you're using Unity - type GParted in Dash should find it
If it's not installed you may install it using Software Center