Food is broken down in the following steps.
Amylase begins breaking down complex carbs in the mouth's saliva
Amylase is neutralized by high acid in the stomach, protein digestion begins due to pepsin and acidic denaturing of proteins into amino acids
Food is churned for a while, then the chyme is pushed into the duodenum. This is were most of the digestion occurs. As a matter of fact, the chyme causes the duodenum to send signals to the stomach to ease up while it processes the current load.
Amylase from the pancreas finishes breaking down the carbs in your small intestine, while at the same time, proteins are digested and fats begin and end their digestion. All at the same time.
Here is where they diverge. Proteins and carbs are broken down enough to pass through the intestinal wall and into the blood stream (the hepatic vein takes them to the liver now)
Fats are hydrophobic, dont like this option, and are shunted into the lymphatic system before distribution.
There you have it. This is how the body digests things. Fat takes a little longer to break down and causes signals to be sent to the rest of the CNS saying "no more food". It doesn't actually hinder the digestion of what was eaten.
-Information given is a paraphrase of what can be found in Human Physiology from Stuart Fox