Instead of transferring all of / to another parititon, if you have like /usr on the same partition, it might be easier for you to move its contents over to the new parition to save space. The majority of the time /usr takes up the most space as that is where most packages will get installed to.
Also check your /var for logs that are getting rather large and you can clean those out along with /tmp too with any temp files that can be removed.
I would suggest to just find the directory that is taking up the most space and move that one, then you will have two or more partitions for you system, etc.
If you want to find an easy way to move data to another partition, check out my latest journal feature, I recently installed a new drive for my /var partition and told how I accomplished this.
PS. Journal button should be below each of my posts.