A lot of the childhood illnesses are not serious if you get them as a child, but are serious as an adult - rubella can cause a type of arthritis for example, if you get it as an adult, and makes you very ill with severe flu-like symptoms, in addition to the standard rash (when I got it as a kid, I got 2 weeks of school for being infectious, whilst feeling perfectly healthy), so there are good reasons to have a vaccination against childhood illnesses if you’ve never had that illness.
Mumps, IIRC, can cause sterility in males, so that’s another reason to vaccinate against it.
Pertussis is whooping cough, a fairly nasty disease, but rarely caught outside of childhood & AFAIK no more or less serious at either age.
As for how the vaccines work, it depends on the disease - some vaccines contain live, but weakened forms of the virus/bacteria, some contain dead forms of the virus/bacteria, and some contain fake forms of the virus/bacteria - it looks like it to your body’s immune system, but it’s not the real virus, dead or alive.
The trick here is that your body fights viruses & other diseases by producing chemicals/cells that lock onto the outside of the virus, and destroy it. To protect you against a new disease, there’s a time lag while they work out the shape they’re going to lock onto & start to produce the right antibodies. During that timelag, of course, the virus is reproducing like crazy & making you ill - in some cases killing you.
If they give you a weakened form of the virus or bacterium that causes the disease, or a shot of dead virus/bacteria, your body will get to learn the shape to react to without having the disease caused. Then if & when you get exposed to the disease, your body can just start whacking out antibodies straightaway with no learning period (& it usually maintains a low level of the antibodies constantly too, so it should mop up the odd one or two viruses that happen to occur before they even need that wholescale warfare gearing up to happen).
You do sometimes get a less severe form of the disease because you’ve had the vaccine - the MMR vaccine apparently can cause a mild fever and a rash about 3 weeks after the shot (that’s about the usual incubation time for the measles virus for the symptoms to show) and that’s because it’s a live virus, but a weakened one that you get given.
Now then - what diseases haven’t I covered?
Polio - this disease can kill you, or leave you paralysed, at any age. There’s a new combined vaccine out for Diptheria, tetanus & polio that allegedly covers you for life. If you haven’t had it, get it. If it’s not available where you are, get your booster shots ever ten years.
Tetanus - this is a bacterium, the spores are found in soil, so it’s all around you, theoretically, but can only enter the body through an open wound - any time you get a bad cut that needs stitching, they’ll usually ask you at the hospital when your last tetanus shot was. If it’s over 5 years, they tend to jab you again, so you feel doubly in pain ;). It causes a disease called lockjaw in its common name - the bacteria produce a toxin that causes all your muscles to try to contract. All of them, at the same time. Causes pain, paralysis & death (heart can’t keep beating, lungs can’t expand).
Diptheria - caused by a bacteria, and again, it’s the toxins it produces that kill you - they can damage heart & nervous tissues. In the UK, in 1940, 46,281 cases and 2,480 deaths were notified, but since vaccination came in, only 2 deaths have been recorded since 1972 & most cases are acquired abroad by people who have let their vaccinations lapse.
Hepatitis you definitely don’t want - it can be transmitted by blood or by sexual conact - ther are various different strains & they recomend different vaccines depending on where you’re going - Hep A & B are the common ones to get protected against, but it varies by country of destination. You can also get a vaccination of antibodies against the disease if you leave it to close to your trip to be immune from the actual proper vaccination. The immunity from that only lasts about 3-6 months though, and the immunity from the other method lasts about 3 years for the current vaccines.
The antibodies are prepared from human blood (Immunuglobulin A) & the shot is given into the gluteus maximus (backside). If you are at all concerned about health risks from blood products, or have religious reasons not to receive blood from other people, then you have to skip that method & have the proper vaccine (which is a much better option anyway - longer protection).
I don’t know anything about that type of influenza, but I do know that people at risk are often vaccinated against whichever the current strain of flu is - usually the elderly & the very young, and anyone with a weakened immune system, or possible complications from respiratory diseases, like asthma. That needs doing each year, as the virus mutates so fast & changes its outer coat, so your body stops recognising it.
I think that covered everything. If you have more questions, just ask.