Axolotls have four different forms of respiratory systems, cutaneous, buccal, lungs, and gills.
The most obvious and common is the gills. Now since this is the axolotl, they're already going to be a little odd with their external gills, the external gills are actually membranes of capillaries, that branched out from different arteries. The water seeps through the gills and filters the oxygen into the bloodstream, then passes the carbon dioxide through the skin. Also, unlike the typical gills where the organism has to move in order to filter the water, all the axolotl needs to do is wiggle their gills.
Cutaneous respiration is simply breathing through the skin. The axolotls have very delicate and thin skin, thus they are able to absorb oxygen straight through their skin.
Buccal respiration is done through the buccopharyngeal membrane which is at the back of the throat. The membrane flaps, allowing oxygenated water to filter through.
As a last resort, or just because they want to, the axolotls can use their lungs. Axolotls will swallow air bubbles and even go up to the surface for air, and then burp out the access air later on.
The axolotl has what is called a three-chambered heart. Meaning there are two atria and one ventricle. Now, this can be dangerous as the non-oxygenated blood and the oxygenated are at risk of mixing, however, the timing of the atriums contractions is what keeps them separated. The technical term for this is an incomplete double circulatory system.
The digestive system starts with the mouth and the buccopharyngeal membrane, once the food has passed the buccopharyngeal membrane it then goes into the esophagus and passes through into the stomach, which then goes into the small intestines, where the nutrients are absorbed. Now the axolotl doesn't have a large intestine, so the waste goes into the rectum, but not the anus as the axolotl doesn't have one, instead, the waste will exit out the cloaca.