It might be possible to upgrade the system but you’ll probably need to find someone who knows these systems better. There used to be a cover on this panel where the buttons and speaker are; that might’ve given a make or model number, or maybe even had a sticker for the company that installed or monitored the system. This is the control panel but there might be a hub/brain somewhere else where all the wires lead, perhaps in a closet or utility room. You might find more information there.

The system sounds like it’s functional; in most jurisdictions the sellers should have provided you with the information necessary to use the system, like alarm codes and contacts for a monitoring service (if any). If you get a code that might be all you need to use the system (you should change the previous owner’s code).

It’s possible to use the alarm without a monitoring service, but I feel like they’re more useful with a service. If there is a monitoring service they would be able to have someone come check out the system and reset the codes for you, show you how to use the system. Is there a sign in front of your house with the name of an alarm company?

This looks like a system old enough that it would probably need an actual telephone line for monitoring and uses wired sensors. Wired sensors are probably a good thing. The sensors don’t need batteries and aren’t vulnerable to jamming like wireless sensors. They’re more expensive to install, though, especially in a finished home. Since yours is already installed, that’s a boon. Most current systems are probably designed for wireless but I’m sure there there are ways to use the hardwired system with more modern systems and get features like remote access.