You're both experiencing 'delayed-onset muscle soreness' (DOMS). I personally tend to feel it most 2 days after a workout. However, the more you go to the gym the less you'll experience it. It'll come back if you take a break from the gym, start doing different exercises, or increase the intensity of your workout.
With the deadlift, it's a full compound exercise so you'll feel it in a lot of muscle groups. Just make sure your form is good, have someone check it for you. I know that my form starts to go when I fatigue, so I have to be more conscious. Not sure why you'd feel it in your chest, might be from another exercise.