How does the encryption/decryption process work? I am not referring here to the process of Public Key exchange, but rather the process of encryption or decryption itself after the Public/Private keys have been generated. For instance, using Diffie-Hellman algorithm we ended up getting both Public and Private Keys of size 512 Bytes. How are we supposed then to use the Public Key to encrypt a given plain text message "Hello World!" and the Private Key to decrypt it?
Diffie-Hellman does not generate public and private keys. It generates an agreed number in such a way that:


An eavesdropper cannot work out what number has been agreed upon.
A man in the middle can either know the agreed number which party $A$ is using and the agreed number which party $B$ is using, or ensure that the agreed number which party $A$ is using is the same as the agreed number which party $B$ is using, but not both.


No encryption is involved in Diffie-Hellman and Diffie-Hellman does not generate encryption keys.

Of course one of the main uses of an agreed number would be to do encryption with it – or rather, with a key derived from it. But that is nothing to do with Diffie-Hellman per se.

To that extent, your question is "How, given a number, do I turn it into an encryption key and use it?" – to which the answer is, "In any way you like".