Work PC is slow and the fan gets loud

One of the Windows PCs I use at work has gotten really slow lately, and the fan spins up loud even when I'm just in a browser and email. It's a small desktop, not a laptop. I'm in IT support so people hand me stuff like this, but I honestly don't know how to tell what's causing it. I've only restarted it and closed some programs, and the fan still ramps up after a few minutes.

I think the first thing to check is dust, but I haven't opened it yet. Is there a simple way to see if it's overheating without special tools, like a free temperature program or just looking at Task Manager? And how do I tell if it's a hardware problem versus Windows just doing something in the background? What would you check first?