Do I have to upgrade any of my other hardware?
Your motherboard needs to have the right kind of slot to accommodate the card.
There needs to be enough physical space inside your case for the card to fit. Cards are trending pretty large these days.
Your power supply needs to have enough wattage to power the card, as well as everything else in your computer. You can find calculators online that allow you to pick specific components, and it will add everything up to make sure you have enough power. It is reasonable to have a bit more than you need.
Most good graphics cards these day require additional power, plugging them into the motherboard is not enough. This may be a 12 volt power cord coming from your power supply, or it may be more than one power cord coming form your power supply. Make sure the power supply you have has the correct wires and plugs for the type of graphics card you want.
Keep in mind the type of video outputs the card has. I picked up an AMD R290, and it has two DVI outputs, one HDMI output, and one DisplayPort output. I can drive my two old 1080p monitors, my television, and my new 4K monitor all at the same time with this setup. Consider not only the screen(s) you are using now, but what screen you might get in the future.
That's all I've got; I'll let someone else answer the rest of your questions.