This type of shot is called a combination shot. In addition to combination shots the recommended schedule is designed so that children get more than one shot at the same time. When shots are given like this, it is called simultaneous administration. The shots recommended by the CDC protect your child from suffering major problems, and possibly death, from the diseases they are made to prevent. The earlier your child receives the shot, the lesser chance of your child suffering problems from the disease.
Vaccines are very safe, and there are always safety checks going on to make sure they stay safe.
Serious side effects are rare. Babies experience many illnesses, small and large, while growing up. It's very common for babies to have sniffles, ear aches, and colds. When a shot happens just a couple of days before a baby gets sick, sometimes parents think there's a connection. This idea is called temporal association and it means that the only connection between a shot and an illness is time.