df shows you available, remaining, and total space on mounted partitions. the -h option gives it you in a "nice human readable" format using K, M, etc as appropriate. The -m and -k options will instead show it to you as number of 1K or 1M blocks.
Use du to see how much space is used by a directory and its subdirectories.
Running fdisk -l as root will show show the sizes of partitions on your disks and any unused space. There is likely one or more other partitioning tools in your distro that will also give you this info.
On all of these (and most other commands), consult the man pages for more info, including all of the options and what they do. For example, to find out info about df, type:
Your distro/DE may also have graphical tools for some or all of this. If you wish somebody to post about that info, you should post what distro, and if possible, what desktop environment (KDE, Gnome, etc) you are using.