Learn Hardcore Russian: Я не люблю пластик.

Russian often does drop subject pronouns when the subject is clear from the verb ending, but using я is perfectly normal and quite common, especially:

when you’re stating an opinion or preference
when you want to emphasize I (as opposed to someone else)
in short stand‑alone sentences

So both are possible:

Я не люблю пластик. – Neutral, clear, maybe a bit more emphatic: I don’t like plastic.
Не люблю пластик. – More casual, like Don’t like plastic (myself). Often in conversation when context is clear.

Both are correct; including я here sounds very natural.

Why is the negation не before the verb люблю? Could it ever come after it?

In standard Russian, the basic rule is:

не + verb

So:

не люблю = do not like

Putting не after the verb (люблю не пластик) no longer negates the verb itself. Instead, it shifts the meaning to contrast the object:

Я люблю не пластик, а стекло.
I like not plastic, but glass.

So:

Я не люблю пластик. – Normal negation of liking.
Я люблю не пластик. – Means I like something other than plastic (and you must say what that is, or at least imply it).