In short, oxygen travels over large distances in the body by binding to hemoglobin, which is located in red blood cells. Oxygen can also be stored in myoglobin in the muscles, and will travel over short distances by simple diffusion.
Carbon dioxide travels in the blood largely as bicarbonate ion. Inside of the red blood cells there is an enzyme called carbonic anhydrase that catalyzes the reaction CO ₂ + H₂O <--> H⁺ + HCO₃⁻ (and also <--> 2H⁺ + CO₃²⁻😉. Most of the carbon dioxide travels in red blood cells as bicarbonate ion. Some bicarbonate ion will stay be in the plasma, but most isn't. Some carbon dioxide can also bind to hemoglobin (not to the same sites as oxygen, however) and travel through the body bound to hemoglobin.