Strictly speaking, you will never get them to match exactly (for a lot of reasons) and even pros have an industry devoted to trying to get printed output to match the designer’s intentions and more often than not they have to make changes based on the proofs.
But that’s probably not necessary.
I don’t know about Elements, but PS proper has a color management feature that will go a long way to making things better. Open Control Panel and if you have Adobe Gamma in there, run the wizard. You will adjust your monitor settings and Elements should be able to figure out what things look like on your monitor and then save that “color profile” for when you print and the colors will be truer. *
This becomes important especially when sending something to another printer (like a pro press) where you download their color profile so the program will adjust the colors so they appear the same for you as they would if you were working on their machines.
The $300 program is probably this, but it comes with a sensor that attaches to the screen and calibrates it much more accurately than simply adjusting the brightness.
I’m not sure how or even if the program recognizes what your monitor displays, but it does work.
When we had our new color printer (a digital laser copier, actually) and its accompanying Fiery server installed at work, the technician explained to us that the blues would not look the same on the screen as when printed, and that there was nothing to be done about that without a huge amount of work and color-matching, and that even then there were some shades of blue that printers, either laser or inkjet, are simply incapable of printing. The problem isn’t limited to blues, of course, but it’s blues that make it most obvious. He had a little graph showing the gamuts of visual color, color that computer monitors are capable of displaying, and color that can be printed by various types of printer. Each of those areas were successively smaller. (It was kind of like an amorphous Venn diagram.) I wish I had it handy.