This ...
but I have no external way of backing up my files (
makes me answer this ...
Is there a way to install Ubuntu and have it run smoothly and still keep my files?
with no.
If it is your 1st time trying to install Ubuntu you will run into problems. You will break things. You will make the wrong decision at some point in the installation. Heck, even I myself have set up tons of operating systems and make mistakes where I lost a partition or data that I did not want to remove. Backups are the safety measure you have to take to be relatively more certain you can get your files back into an operating system.
To make the answer be "yes" you must create backups and store your files on something where discissions made while installing an operating system or messing with partitions, disks or anything a level lower than the operating system itself.