What is the best thing to do for a bad muscle strain, or extreme muscle soreness?

Rest/Fluids/hydrate. Along with rest, it's important to drink lots of fluids and stay well hydrated when recouping from a strain or sore muscles. Being dehydrated will make it harder for your body to flush out the Lactic Acid and other chemicals that cause the soreness. Being well hydrated will also make it easier for your body to heal more quickly on its own.
Rest. The most important thing to do initially is to rest the area affected. Icing and/or heat can be helpful and is a good secondary thing to do. Ultimately, strengthening is important to prevent the strain in the future.