Check health of my system such as database connections, mainframe connections, messaging services etc

I am so new to this topic that I am not even sure I am asking it correctly but I will give my best. We have number of applications talking to databases such as Oracle and Mainframe and uses messaging services.

I wonder if there is some tool or a script that could be written or something that would allow me as a manager to automatically or manually run it in order to check that:

1. I can connect to all mainframe transactions
2. I can connect to all databases
3. I can connect to all messaging services
4. I can connect to middleware

As described, I dont even know where to start and how to start and I was wondering if someone had to do similar to this and can provide some helpful insights.

If this is wrong place to ask such question, what is the right place to do so?