Latex can be harvested directly from trees, and still is, but most plastics used today are from petroleum.

Simply put, plastics are chains of carbon+other stuff. Petroleum is very rich in carbon+other stuff. Refineries break down big chains of petroleum into smaller parts. They then combine these smaller parts into other big chains, which are the various types of plastic we use today. The same can be done with natural latex, but that stuff isn't as abundant so we don't use it much anymore.

An analogy of this would be starting with a house. You break down the house into smaller parts: wood, nails, drywall, paint, and so on. You then use these parts to build a bunch of sheds.

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Plastics are created from chemicals by reacting them together. If you have a particular plastic in mind you can find very cool videosof people doing it.

Where we get these chemicals depends. That said, oil is often a cheap and convenient source for them.

Plastic is created from a monomer. Think of them as links in a chain. The classic example is ethylene. Scientists have discovered that treating it you can create chains of them to get polyethylene. This pattern repeats for different plastics. The feedstock can be oil or natural gas. But they have to go through further refinement before they can be turned into plastic. When I was in chemistry, the trick was to start with a pure feedstock. Otherwise you run into having to deal with side effects & impurities or may end up with a lower grade of plastic.

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