Weightlifting gloves improve your grip on weights and make workouts safer. Hands get sweaty during a workout. When they do, your hands can slip. This can be annoying if you are doing pull-ups and keep falling off the bar. It is also dangerous if you are holding heavy weights over your body.
This is always a bit alarming, but it's perfectly normal. When you first start working out, your body responds by raising your heart rate and causing you to breathe heavy. Exercise gets a lot easier as you go along, and it's because your body starts adapting to your workout.