how to upgrade my Hard drive on MacBook pro
I thought that maybe buying a SSD would speed things up. Is this true?


The simple answer is yes. It sounds as though you may also be paging out due to RAM limitations as well, however. When you run out of RAM, your machine uses your drive as a  disk cache, essentially, which slows down your machine ( a HDD/SSD is meant to be a dedicated storage device, not temporary). That being said, upgrading to a SSD would still make the machine quicker even if you are paging out. 


  If So will I be able to back up my partition when upgrading? What would be the best way to go about it?


Yes, you will be able to back up the other partition. I highly recommend following this clear and concise guide, by ASC member clintonfrombirmingham when doing so.