Do I really need to buy software just for backing up my OS before swapping out my HDD for a ssd?

You need some free cloning software such as macrium reflect and a hard drive caddy to temporarily house the new hard drive. This will create an identical copy of the hard drive onto the SSD. You may be able to use your current external drive to store an image file of the HDD clone and then restore it to the SSD, you would need to check if macrium supports this
http://www.macrium.com/reflectfree.aspx

google 2.5 external hard drive enclosure

You have no need to buy software for backing up your OS. Because there is a free backup software -AOMEI Backupper can help you backup OS in an efficient way. But something need to mentioned is that do not keep the backup on the same drive, you need another storage to store your backup.

You back your files up to your external hard drive, yes. But how you do that is up to you. You can buy software to do it, you can do it manually (edit/copy, edit/paste) or you can install a free program.

The best FREE backup program I've ever used is SynToy. It's made by Microsoft. I use it on both of my home computers and I back up my files to an external hard drive using it. In this scenario, you are backup your personal files. But any software on your computer will have to be reinstalled. Here is the link:

https://www.microsoft.com/en-us/download/details.aspx?id=15155

Now if you want to "take a picture", so to speak, of your entire hard drive, and then put that "picture" on your new hard drive, you can also do that. For that, you will need some software to clone your hard drive, or "take a picture" of it. Then install that picture on your new hard drive. I use Norton Ghost, but it's not free, however you can use their free trial. That is here: http://buy.symantec.com/estore/clp/productdetails/pk/ghost-solution-suite

Backup of an OS is normally a total waste. You need to re-install on the new hard drive to ensure the drivers are correctly present.
Backing up your DATA is a good thing - but a copy using File Explorer is all that is needed.

You can transfer the files, but you will need a fresh install of windows for the new SSD.

Yes... unless your going to write a program/software to do it yourself.

I Use Acronis.

You can not just back it up and copy it. You need to clone the old drive to the new one.