For example, which of these is correct:
Ich esse gern Pizza
Ich esse Pizza gern
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Both are correct. To me, Ich esse gerne Pizza hints at a question like Was ist Dein Lieblingsessen, whereas Ich esse Pizza gern would be more appropriate as an answer to Magst Du Pizza? That is, I would put the important word (the one carrying new information) at the end of the phrase.
Ich esse gern Pizza sounds more natural than Ich esse Pizza gern to a native speaker, even though both are correct.
Better: Ich esse gerne Pizza (standard language)
It depends. When you want to express that pizza is your favourite meal you can say:
Ich esse gern Pizza.
When you admit that you would be happy to eat pizza, e.g. that has already been ordered or served, you would say:
Ich esse Pizza gern.
In the first case, I would prefer "gerne" instead of "gern".
You can also say
"Ich mag Pizza" - this means that you like pizza.
"Ich esse Pizza gern" is a more formal way of saying it but if you want to say it to friends you should just say "ich mag Pizza" - they all mean the same thing.
The "proper position" depends of emphasis:
1) Ich esse gerne Pizza means I like pizza. (As opposed to hamburgers or frankfurters.)
2) Ich esse Pizza gern means "I like pizza." (As opposed to hate it.)
In either case, the part to be emphasized is at the end of the sentence.
Between the above, 1) is considered a bit more "standard" than 2)