I started out iOS development 4 years ago when there was only two screen sizes to worry about. The 3.5" (non-retina) iPhone and the 9.7" iPad. One of the reasons that I chose iOS development was because unlike Android OS development at the time, there was only two screen sizes to design for making it simple to ensure that all my apps would look good on all devices.
I am the kind of person that likes to make sure everything is perfect before releasing my application. So, I believe that if you've only tested your app out on the simulator (of which there was only like 4 options, two versions iOS for the iPad and 2 for the iPhone) then your app is not ready for the big time because code runs differently on the actual device itself. So, four years ago, I got an iPad and began to test my applications on that as well.
However, this is where the issue (and the question) really comes in. Now theres 2 or three different size iPhones and like 5 different size iPads. I can't very well buy one of each (nor do I want to) so what do you guys recommend would be the best way to test my apps on-device for all these various screen sizes?
Buy used devices on eBay. Buy devices with small damages. iPhone 6 with cracked screen is quite affordable on eBay :-) For iPads, don't worry about the mini, it is just a shrunk version of the larger one and behaves identically (alternatively, buy the iPad Mini and don't worry about the larger one, saving money).
Use layout constraints, and use the simulator. If you really don't want to release unless it is tested on a real device, just don't support the bigger sizes. Your code can easily treat a 6 or 6+ as a "large iPhone 5".