Some people think vaccines are safe to use, and they are completely correct.
The goal of vaccines is to create immunity to certain diseases. The first vaccine was created by Edward Jenner in 1796. This stemmed from a simple discovery of his, handmaids that had caught cowpox would never contract smallpox. Cowpox isn’t really dangerous, while smallpox often kills. He tested this on an eight year old boy by first giving him cowpox and a few weeks later giving him fluid from somebody with smallpox. They boy did not get smallpox, so morals aside, it was success. 
How the Immune System Works
To first understand how vaccines work, you must understand the problem they are helping to solve.
Bacteria and viruses love to get into our body, in fact we have more bacterial cells in our body than human ones. This means that inevitably, some will harm us in some way. These organisms, called pathogens, are encountered everyday. Luckily body has developed a system to help combat these unwanted pathogens. This system is called the immune system. 
The Immune system is run by different types of white blood cells. The first step involved the Macrophage. The Macrophage engulfs any cell or debris it finds that does not have specific proteins pertaining to healthy human cells on its surface. After engulfing the cell left over pieces called antigens are leftover. These antigens are identified as dangerous and the body then attacks them using other cells.
Antibodies are just as important, it's almost impossible to stop infections without them. They are created by cells called B-cells. These B-cells are either dedicated to creating antibodies, or dedicated to remembering antigens for future use. Antibodies come in many different forms, but they all have one purpose. This purpose is to attach to specific proteins. These proteins (almost always) belong to pathogens. This allows pathogens to be easily noticed by other white blood cells. Without these antibodies many types of pathogens can go unnoticed throughout your body. Antibodies also inhibit the pathogens by blocking interaction with other cells and slowing/stopping movement.
We also have T-lymphocytes, sometimes called T-cells. These cells are designed to attack infected human cells. The T-cells come in two different variates, a helper T-cell and a killer T-cell. The helper “scans” cells to determine if they are infected, and if they are the Killer T-cells are alerted and kill the specified cell. These cells are responsible for most autoimmune disorders. 
The Need for Vaccines
The immune system works well to stop any pathogen that has been encountered before, but if it hasn't there likely be no antigens made. With no antigens there will be no antibodies. With no antibodies, a disease can roam your body freely. It often takes days to build up any sort of immunity. This window of opportunity can be deadly. This puts us in a tricky situation, we can’t expose ourselves to a pathogen with permanent effects, but we can’t protect ourselves from it without exposing ourselves to it.
Vaccines are a way around that. Exposing ourselves to a weakened pathogen or another one similar enough, can still give us an immunity. That doesn’t happen safely often in the wild, so we must do it ourselves.
There are multiple types of vaccines, each is equipped to fight a specific set of pathogens.
These vaccines are specialized in the immunization of viral diseases. This type of vaccine involves putting a “live”, but weakened virus inside the body. This usually is no trouble for the body to fight off, but in some cases where a person has a weakened immune system this type of vaccine should be avoided. This type of vaccine is used for: measles, mumps, rubella vaccine (MMR) and varicella (chickenpox).
Inactivated vaccines use “dead” viruses. This type is often used for more dangerous viruses. It often takes multiple doses to build up an immunity, because the immune response is weaker with dead pathogens. This type is most commonly used with Polio.
Toxoid vaccines are designed to provide immunity from bacteria that produce toxins. These toxins are weakened so the immune system can learn to deal with them. One example of this is the DTaP vaccine, which contains diphtheria and tetanus toxoids.
Subunit vaccines use only parts of the virus or bacteria. The only parts contained are the ones needed for the body to create the correct antigens. This decreases the chance of any side effects. One example of this vaccine is the pertussis (whooping cough) component of the DTaP vaccine.
Conjugate vaccines use a different type of bacteria as a substitute. This is so that young children’s immune systems can get used to polysaccharides, a sugar-like coating on the outside of the antigen. This disguises it and the body needs to learn to read it. One example of this type of vaccine is the Haemophilus influenzae type B (Hib) vaccine. 
The other Components of Vaccines
Vaccines do contain chemicals outside of the just the pathogen. Each of these has a reason for being inside the vaccine.
Aluminum comes as either a gel or a salt. It is used to stimulate the immune system so that it responds quicker to the pathogen, increasing the amount of antigens produce and lessening the chance that the pathogen can spread farther than intended. The total amount of aluminum that a baby receives from vaccines is 4 milligrams (4 one thousandths of a gram). In this time period breast milk gives them 40 milligrams, and if soy milk alternative is used 120 milligrams.
Either way the majority of it is taken out by the kidney in less than a day.
Some vaccines contain Antibiotics, to prevent the spread of the pathogen. You get hundreds of times more Antibiotics from drinking water daily. (A ton of that runs off into the environment from farmers and causes problems, and it isn’t even needed.)
Some vaccines are prepared using chicken eggs. This is because the easiest and safest way to get some pathogens to replicate is to use eggs. This includes the influenza and yellow fever vaccines. The amount is small enough that people with allergies to eggs are almost always fine. (Although if you are severely allergic you should do it in the presence of a healthcare provider that can recognize and respond to any severe reactions.) 
Formaldehyde is used to inactivate toxins in toxoid vaccines and to prevent unwanted bacterial and viral growth. Most of it is removed before packaging, but small amounts remain. Formaldehyde is essential in DNA replication and already exists in the human body. About 1.5 times the amount found in any given vaccine is in an infant's body. Animals have been safely given amounts over 600 times that in a vaccine. The DNA replication process is the same, so it is completely safe.
Monosodium glutamate (and sometimes 2-phenoxy-ethanol) are used as stabilizers. They prevent the vaccine from becoming charged when exposed to light or heat. MSG is a sodium salt of glutamic acid, which is a non-essential amino acid. That means our body already makes it. It is also often found in food.
Studies have shown that too much of it is bad for mice, but it turns out mice are just overly sensitive to it. They can only intake about 1.8 grams per kilogram body weight, but over 6 grams a day per kilogram of body weight is considered safe in humans. Either way a cup of breast milk contains over 900 times the amount found in a vaccine and the safe daily intake is over 32 thousand times the amount found in a vaccine.
The last chemical found in a vaccine is Thimerosal. Thimerosal contains a mercury outer coating and is used to prevent vaccine containers with multiple doses from becoming contaminated (single dose containers, which are becoming more common, do not have this). This mercury is both chemically inert, and in extremely small quantities, 1 part in 100,000 to 1 part in 10,000. In your typical vaccine the amount of mercury is equal to the amount in an average 3oz can of tuna. Also there have been an entire body of scientific studies about the effects of thimerosal, it has no measureable effect whatsoever. 
Also it's not really used in modern vaccines anyway.
Vaccines are safe and needed to keep society from getting killed by diseases. I will probably make a follow up post on how herd immunity works, just to cement the idea.
also its not just "some people" its literally every single doctor in the world lmao