Does the Ubuntu Software Center keep a log of everything it has installed? So I could just parse this log to convert it into a script full of 'sudo apt-get instals' or something. And then the next time I install Ubuntu I could just this script to re-install everything.
Yes, It maintains a log in
/var/log/apt/history.log. Older histories gets compressed as
.gz files. It contains installation, upgrade and removal logs.
For keeping existing software backups you can use
dpkg-repack to make
.deb files out of installed files. Here.
If this is for personal use, there is a way to sync programs built in to the software center. If you just click File -> "Sync Between Computers". You just have to create an Ubuntu Software Center account. Here is a how-to with more information which may be relevant, http://www.howtogeek.com/111989/how-to-sync-quickly-reinstall-applications-on-ubuntu/.