Is it a hard drive issue or just Windows 10 acting up?

So, I've got this Dell Inspiron. It used to be fast as hell, but for the last two months, it’s been acting like it's running through molasses. Mostly when I'm browsing. I'll get maybe two pages to load before the third one just gives me some "no connection" error. It's definitely not my Wi-Fi because everything else in the house works perfectly fine. To make matters worse, for the past few days, it won't even boot up. I tried the whole shut down and restart thing, but nothing. My antivirus says there's no malware, so it should be clean. I did upgrade to Windows 10 right around the time things started getting sluggish, so I'm wondering if the OS is the culprit rather than the actual hardware.

Is there any way I can troubleshoot this myself so I don't have to drag it down to a repair shop?

Once you've finished those steps, there might still be some lingering incompatibility issues with Windows 10, so I would suggest attempting a driver update.
If you happen to suspect the drive itself is the culprit, your first move should be to identify exactly what model is installed by using a system information utility such as Speccy or AIDA64; honestly, depending on the specific laptop model, you can likely find that information quite easily just by browsing the Lenovo website.
Furthermore, keep in mind that every major drive manufacturer—be it Seagate, WD, or General Electric—provides its own dedicated diagnostic tool, which you can simply download from their respective websites to run a thorough test.

The Windows 10 start menu is constantly glitching out—happens to me on two out of three machines I work on.

You could try what wiredotter14 suggested.
Also, if you can back up your files, I’d suggest just doing a full factory reset.
Windows 10 has that option right in the settings; you can choose to wipe everything (it claims it'll save your files, but honestly, just make a manual backup first).

My laptop was acting up too, and this fixed it.
Just google Windows 10 Factory reset.

PS,
Even if you end up taking it to a repair shop, you should be backing up your data anyway, so there's really no harm in trying this first.