Lactate (not lactic acid) is produced during intense exercise, because of the large flux through glycolysis (faster ATP production to meet demand). Not all the pyruvate can be accommodated via the citric acid cycle in fast twitch muscles (low oxidative capacity) so most is converted to lactate, which is then taken up by the liver, heart, slow twitch muscles and even the brain to be fully oxidised to carbon dioxide. The pain that you are experiencing during such exercise is most probably due to small lesions that form in muscle fibres, especially in untrained muscle, because not all the muscle fibres work together. Some pain is also due to ischemia (lack of blood flow) and will subside when the muscle is relaxed. The pain and stiffness you experience a day or what after intense exercise, also called delayed onset of muscle soreness (DOMS) is due to inflammation and rigour in some of the depleted muscle fibres.
Just a correction on the fate of lactate: in the liver it will be converted via gluneogenesis to glucose while you are exercising (Cori cycle). During intense exercise your muscle will also produce some alanine from pyruvate via transaminase reaction that will also be utilise by the liver to form glucose. Glucose from glycogenolysis and mobilised via glygenolysis (in both liver and fast twitch or type IIx muscles) will contribute to the supply and high flux through glycolysis leading to massive increase in lactate production within the first 15 minutes of intense exercise. The main limitation of your body is the glucose supply (glycogen stores) and how fast it can remove the lactate and convert it back into glucose. You can improve this type of fitness by intense exercise (20-30 minutes) with ample recovery (24 hour hours) in between. Recovery is aided by low intensity aerobic exercise, which improves blood flow and therefore healing time.