The procedure is called certificate validation and is pretty standard. Some classes and components perform validation for you, others leave it for your manual implementation and control.
Validation ensures (in ideal world) that you are connecting to the legitimate server, i.e. the server whose host name and the name in the presented certificate match. This requires that the server has acquired a valid CA-signed (we omit self-signed variants for lack of security and flexibility) certificate for the needed host name. So far so good.
Now you can either rely on pre-implemented certificate validation or implement your own or add your own checks to the pre-implemented validation procedure. Implementing your own validation is too cumbersome for your task, so let's assume that the client code you use already performs some validation (you have not specified what exactly code you use for connection so I can't comment on it). You can rely on it, however in some countries state agencies perlustrate traffic, and for doing this they acquire (or generate on-the-fly in some cases) certificates which are fake by nature but valid if we follow the validation procedure blindly.
So if you control both the server and the client and also you can implement additional validation (your client component or class lets you do this) then your additional check can be to compare the issuer of the certificate (or the whole certificate chain) to the issuer you know to be valid. This is less flexible and to some extent against the PKI rules, but this approach significantly reduces the chance for the fake certificate to be generated and accepted as valid. The idea is that you know what certificate you use and what CA you used (and maybe use in future), so you can store this information in the client and compare it during validation.
You can read more about certificate validation by simply searching here on SO for "certificate validation" - this is quite popular topic.
