How do I re-size my partitions?

You can not resize a disk if you running from that disc.
Another hint, in the tab in the picture you show it says partition and in the text it says erase so do not use it this way.
If I am right you can resize partitions if you start up from recovery mode and start Disk Utility from there, I think you need to restart and hold the option key to start up from recovery.
Edit: I think I went wrong here because it is still on the same disc, I think you need to make a bootable USB stick or DVD first

Anybody, if I am wrong please say so.
I will try to google for you.

Edit: Found a link but it is for 10.5.x, maybe that is still valid, except yours is newer.

Resize

 

Use SuperDuper! to clone the second partition to an external drive. Then, delete the second partition. Now you can remake the second partition, allocating more space than before. Finally, use SuperDuper! to clone from the external back to the second partition.

 

No need to, you can resize the disk on the fly.

 

I know you can resize disk0s2 (Lion 1 in this case) on the fly, but it was my understanding that it's not possible to resize other partitions without losing data (at least not without third party software like DiskWarrior and the like). Is this not the case?

 

Can Diswarrior do something like that, since when???
Edit: Just looked, NO

I never did but I think if he resizes the first partition to a smaller size he then can resize the second partition larger, I could be wrong though.

 

iPartition.

But it is not free.