Linux, checking how much space is used in each partition

The command-line tool to use is df.

In pretty form, df -h to get the results in human-friendly form.

At the bottom of every tab/window in nautilus it will display how much free space is on the drive you are currently browsing if you have no folders/files selected.

If I'm remembering correctly, Mint is a slightly re-tooled version of Ubuntu. If this doesn't work, you may need to check the repos for Disk Space Analyzer.

You should be able to go to Application > Accessories > Disk Space Analyzer

To see the size of and disk-space used of partitions...

for a pretty, graphical view you can use gparted. If it is installed it will be under:
System -> Administration -> Partition Editor

if it's not installed you can remedy that by using synaptic (I assume mint has that or a similar app since it's based on Ubuntu) or just typing the following command into a terminal:

sudo apt-get install gparted