Flash a Chip

There is no "best" of anything. You select your FPGA or CPLD based on your design requirements... which you haven't said what they are.

I designed a program that will run a gate operator, basically opens and closes a motor depending on different parameter. So in order for it to work I will need to store this logic to a memory device and put it on a circuit board. I originally used a chip that was a stored as .hex file but I am moving on to using this. Is this type of thing plausible? I'm willing to purchase a FPGA or CPLD just want to know the best course of action.

It looks like you have a microcontroller (MCU) programmed with some code and now you want to try to use an FPGA or CPLD? You didn't say what "this" is in "...moving on to this". FPGAs, CPLDs and MCUs can do a lot of stuff. Yes, it is possible to do what you want. Or at least what I understand you're trying to do. It is a matter of learning the tools and the device(s). IMHO FPGAs have a bigger learning curve than MCUs.

So as of right now I have a design in quartus and I simulated it in modelsim and it does seem to function correctly. The next step I wanted to take was to convert it and save it to any kind of chip so I could put use it on a circuit board. I wasn't sure of the different routes to go to accomplish this task. I am not completely familiar with the functions of fpgas but it is something I am more than willing to learn.

What is the target device in your Quartus project? Once your design is compiled you are almost done. Assuming there were no errors during compilation that is. The Quartus software has a programmer interface in it. As long as you buy a development kit that has a blaster programmer built in, all you have to do is program the target device and .... good luck. Don't be surprised or discouraged if things don't work the first few times.

I'm using a BeMicro MAX10 because it is the (IMHO) entry level FPGA that Altera has and it does not require an external configuration ic. It is also pretty cheap: https://www.altera.com/products/fpga/max-series/max-10/design-tools.html