You will find here how to discover which directory or file is using your disk space, and therefore where makes more sence to start erasing files.
This is assuming you have all partitions in one disk, if not, do not run this command at the root level, but at each mount point of every disk you have.
<!--break-->
I will apply this on my own hard disk, I have in my Desktop PC all directories on the same disk.
cd /
sudo du -s * | sort -n
Here is my result.
0 proc
0 sys
4 libsmbios_c
12 mnt
16 lost+found
20 media
36 tmp
184 dev
376 root
4356 bin
9252 etc
12007 boot
12800 sbin
70480 srv
94784 lib
332756 opt
4602136 usr
7439864 var
15018948 home
Now go let's say to /var
cd /var
sudo du -s * | sort -n
And here is my result.
0 mail
4 empty
4 local
4 opt
8 games
8 lock
32 abs
160 run
1564 spool
8956 tmp
48228 log
112820 lib
7268072 cache
As I am using Arch Linux in my Desktop PC /var/cache is one of the biggest directory as there are stored the binaries from every upgrade I made to the system.
You can keep going inside each directory to finally find which is the one causing you running out of space.