In short, sugars are absorbed quicker than proteins and fats because they pass through the stomach quicker and their digestion is simpler.
Sugar can be absorbed through the mouth mucosa when applied as a sublingual gel, as discussed here on Biology SE: Is sugar absorbed into the bloodstream through the walls of the mouth?, but probably in much smaller amounts during usual eating/drinking followed by swallowing.
Not all sugars are absorbed at the same rate.
Glucose and galactose do not need to be digested and can be quickly absorbed in the small intestine via sodium–glucose linked transporters (SGLTs) - sodium acts as a cofactor that stimulates glucose and galactose absorption (Lumen Learning).
Fructose also does not need to be digested but is absorbed much slower than glucose via GLUT5 transporters without the help of sodium (Lumen Learning).
Sucrose needs to be digested in the small intestine with the help of the enzyme sucrase to glucose and fructose, which are then absorbed. Glucose stimulates the absorption of fructose (Food and Agriculture Organization). This means that glucose and fructose from sucrose will be absorbed slower than glucose alone, but faster than just fructose alone.
Lactose needs to be digested to glucose and galactose, which prolongs the time in which they can be absorbed.
The absorption rate of various sugars is reflected in their glycemic index (the rise of blood glucose levels after ingestion): GI for glucose = 100, sucrose = 65, lactose = 46, fructose = 19.
Starch is not digested in the stomach, so it can pass through it quickly, and is then, in the small intestine, quickly digested to glucose with the help of the enzyme amylase. The glucose from plain starch is absorbed almost as quickly as when ingested as glucose alone and faster than fructose, sucrose or lactose. This is evident from high glycemic index of foods made mainly of plain starch: cornflakes (81), instant oats (79), potatoes (78), rice porridge (78), white wheat bread (75).
Foods high in dietary fiber slow down gastric emptying and thus absorption of nutrients (Faseb).
Proteins need to be digested in the stomach by the enzyme pepsin (which slows down gastric emptying) and then further in the small intestine by peptidases into amino acids, which can be absorbed. Slow gastric emptying is the main reason for slow absorption of proteins (Diabetes Care, Experimental Physiology).
Fats are not digested in the stomach but they slow down gastric emptying, so they are absorbed slower than carbohydrates (JCEM, Vivo Colostate).
LIQUID vs SOLID MEALS
Liquid meals pass through the stomach quicker than the solid ones, so glucose from sweetened beverages will be absorbed quicker than from whole fruits (Vivo Colostate, JCEM).