Where your provider's responsibility ends
Internet providers are generally only responsible for the line up to your modem (or ONT) -- everything past that point, including your router, Wi-Fi coverage, and home network setup, is on you. This is exactly why isolating a problem to "which device" matters: a provider's support line can usually diagnose issues up to the modem, but a weak Wi-Fi signal or a router configuration issue is something you will need to troubleshoot or fix yourself.
Renting equipment versus buying your own
Providers typically offer to rent an all-in-one gateway for a small monthly fee, which is convenient but adds up over years of use and can lag behind consumer routers in Wi-Fi performance and features. Buying your own modem (compatible with your provider) and router outright, when the provider allows it, is usually cheaper in the long run and gives you more control over settings, though it does mean troubleshooting equipment issues yourself.
Frequently Asked Questions
Is the all-in-one equipment from my provider good enough?
For typical home use, an all-in-one gateway from your provider is often perfectly adequate. If Wi-Fi speed or coverage disappoints, switching it into bridge mode and connecting your own higher-performance router is a common way to improve it without changing your internet plan.
Can I just buy a router and skip the modem?
No. A router has no way to interpret the raw signal coming in on your provider line, so without a modem (or an ONT for fiber), a router alone cannot get you online at all.