Liquids are different than solid materials both in the real world and in Factorio. In Factorio they can only exist in pipes, some buildings and in barrels. Because of this the player can't carry or pick up liquids, Inserters can't interact with the liquid and it can't be dropped on the ground like other items. Now if the liquid is in a barrel then it can obviously be moved or dropped like normal since it is in a solid material/item. However, in the vanilla game the only liquid that can be put into barrels is Crude Oil. Another very important thing to note about liquids when they travel through pipes is that you should never let pipes that are carrying different types of liquid connect. If they do, the liquids will mix and clog up anything farther down the line that needs only one of those liquids. The last important aspect of transporting fluids is that they have a speed limit when going through pipes. Essentially, the longer the pipe is the less the capacity and speed is.
Now that you have a general idea of how liquids work let's go through the steps of Oil Processing. This is actually a large part of the game and one of the most important parts. In order to get started you will need to actually get some Crude Oil. If you pull up the map (M) the Oil Wells will appear as pink dots on the map. To actually extract the Oil from them you will need to use a Pumpjack and place it on top of the Oil Well. Normally when you go to place a Mining Drill down on a resource patch it will tell you how many resources are under that particular mining drill. The Oil Wells work a little differently so I'll explain it below.
When you mouse over an Oil Well or place a Pumpjack over it you will see a percentage displayed on the tooltip window that pops up. Let's say for example you have one that is 180%. Pretty much just divide the percentage by 100 to get how much the Well will produce per second. So for this example, you can pump 1.8 oil per second from the Well. However, over time the production of the Oil Well will decrease all the way down to a minimum of 10% which would be .1 per second. This may seem quite bad, but it actually means that the Oil Wells are infinite because they will never go below this amount. You can also squeeze more out of it at this point by adding Speed Modules or Productivity Modules (these will be explained in a later section).
Once you have some Oil secured, it is time to start actually processing it. In order to do this you will need an Oil Refinery and some Chemical Plants. I would highly suggest finding a big open area in your base to do the Oil Processing because it takes quite a lot of room if you want it to be even slightly organized. When you place down an Oil Refinery and go to choose a recipe in it there are only two possible things it can ever do. The first is Basic Oil Processing, and the second is Advanced Oil Processing (this part will be covered a little later). You will have to choose the basic one to start with since you won't have Advanced Oil Processing Researched yet.
However if you store it in the tanks then the Refineries can keep working even if you aren't using the products as quickly as you are producing them. As mentioned above you need to make sure not to let the pipes of each liquid connect because they will get all mixed together and mess everything up. The pipes will automatically connect if you place them right next to each other, so you need to make sure there is at least one space between them. This doesn't apply for underground pipes though, so you can use them very effectively here. Here is an example of one way to run the three liquids from the Refineries to the Storage Tanks without mixing them:
It is good to note that if the Refineries were not placed at least one or more spaces apart this would not be possible.
Now that you have the liquids in the tanks it is time to actually use them to make other products. All things made from the Heavy Oil, Light Oil, and Petroleum will be made in a Chemical Plant. For most of these it is actually pretty self-explanatory because once you select the recipe you want in the Chemical Plant it will show you were the inputs need to go and where it outputs if the finished product is a liquid. There are a few things that don't stay liquids though. For example, you make Plastic and Sulfur from Petroleum, but both the Plastic and Sulfur are obviously not liquids. When the product is not a liquid, just simply take it out of the Chemical Plant with an Inserter as if it was an Assembling Machine. One other thing to note is that some things actually require solid materials as well, for example Batteries also require Iron and Copper Plate in addition to Sulfuric Acid. Here are a few pictures to demonstrate the process of creating some of these products.
If you don't know what some of the products/items are you can look in the Item Index and each item has a short description of what it is and might be used for.
Advanced Oil Processing
Advanced Oil Processing is the second recipe you can choose in an Oil Refinery, and caters more to the late game item requirements. Advanced Oil Processing requires water in addition to Crude Oil, and gives you less Heavy Oil, more Light Oil, and more Petroleum than Basic Oil Processing. It also allows the Cracking of Heavy Oil into Light Oil, and Light Oil into Petroleum. When you switch over to this from the Basic Oil Processing, one of the inputs on the Oil Refineries will change to water instead of both of them being Crude Oil. To the left is a picture of what it looks like and one way to connect both water and Oil to it.
Luckily the output slots don't change so everything you already have connected there should work fine. The reason that Advanced Oil Processing gives you more Petroleum and Light Oil is because these (primarily Petroleum) are needed more in the late game. This is also why you can Crack the Heavy to Light Oil, and Light Oil to Petroleum. The processing of Cracking takes place in a Chemical Plant and requires water as well as the liquid being Cracked. You do lose a bit when Cracking something because it gives you one less of the product than what you put in. So to word that differently, it takes four Heavy Oil and some water to Crack into three Light Oil, and it takes three Light Oil and some water to Crack that into two Petroleum. So you do lose a bit when Cracking, but in the late game you need a lot of Petroleum and don't have much use for Heavy oil and Light oil aside from Lubricant (made from the Heavy Oil), so it can be worth it to Crack Heavy or Light Oil. For a visual example here are some pictures demonstrating the process of Cracking.
That pretty much covers it for Oil Processing. To recap a few things; Remember to not let pipes carrying different liquids connect because they will mix together and bog up the system. Also remember that some of the items created through the Oil Industry also require solids such as Iron, or Copper or Coal, so if something doesn't seem to be working make sure to check the recipe and be sure you aren't forgetting to Insert or export something.
If you aren't sure what one of the products is used for or how to get it, make sure to check the Item Index and see what it says about the item.