Gentlemen
Men would begin their clothing regiment by putting on their drawers. These were usually simple and made from thin linen. They resembled elongated boxers. Next, a gent might fancy to put his socks on which would be secured at the knee with a garter (a simple ribbon). A shirt that was about knee-length would be tucked into the breeches and between the legs. Voila! Now you are ready to put your clothes on!
Ladies
Of course undergarments for ladies were a more complicated matter. They served the essential function of shaping women in the fashionable mold that God failed to.
The core piece of ladies' small clothes was the chemise or shift. This was a simple white gown that looks like a night gown. Nothing would be worn underneath the shift. It was thought that the ladies' region needed to be aired out. This made that special time of the month a pretty sloppy event since women would just bleed into their layers. At the time of her death, Marie Antoinette had been wearing the same ragged clothing in prison and her uterus was hemorrhaging (poor thing!). In an effort to appear clean and decent, the doomed queen had saved a new, unworn chemise in her cell, specifically for the day she would be brought to the guillotine.
Petticoats would be put on over the chemise. These were skirts, usually two or more, that were meant to keep the lady warm and the gown flouncy.
Stays, or the corset, would be put on over the shift and tightened to give the torso a nice V shape.
A pocket would be tied around the waist. These could be embroidered with pretty designs even thought they were never seen once clothes were on. They were large enough to hold all the necessities and the modern purse is actually an evolved form of a pocket! But that's a whole 'nother post for later.
Panniers would sit over the pocket on the hips to give the appearance of a slender waist. These came in many shapes and sizes, from a cage-like structure to small pads to flair out the skirt slightly. At the end of the century, false bottom (or bustles) and even false stomachs (a fad Georgiana invented) would replace panniers.
As soon as stockings were put on and secured with garters (ribbons again) a woman was considered...undressed. Well, in a state of undress. She was now prepared to put on her gown!
For good visuals, Dangerous Liasons, Amadeus, and The Duchess show the undergarment portion of the clothing process.