Just for curiosity, I am a diabetic and I was wondering if humans would suffer from the same ilness if they were on other planets? Does the climate on other planets help the immune system? Will there be some new health problems?
Human diseases come from many causes.
Intrinsic: Some diseases are due to failures of the human body: mutations (probably most cancers), many auto-immune diseases (including diabetes, alas!), and diseases stemming from old injuries. There's no reason to think they'll be any different off-Earth.
Environmental: Cancers caused by radiation, poisoning due to effects of the environment, sunburn, etc. will go up or down depending on the local environment. It's easy to imagine habitable planets where nonetheless some of these diseases are more common. (UV and radiation in general are expected to be more of an issue on Mars, for instance.) Differences in gravity have already been shown to cause developmental changes that can alter health.
Bacterial and viral: What happens here depends on what we take along with us. We have wiped out smallpox and are close to wiping out some other diseases. It's reasonable to expect that quite a few pathogens can be prevented from emigrating to new planets along with people.
Zoonotic: Some diseases -- possibly most diseases if you go back far enough -- originally jumped to humans from animals where they are endemic in the wild population. (Ebola, for instance, almost certainly comes from an animal reservoir. HIV jumped to humans fifty or seventy-five years ago. Influenza may have come to us from birds.) It seems highly unlikely that we'll bring wild populations with us without eliminating any diseases that they carry. (Eventually, new diseases will evolved of course, but not quickly.)
Alien: It seems very unlikely that alien diseases will affect us -- our biologies will be too different -- but we can confidently expect that if their biology is at all like ours we'll be allergic to a great many of their proteins.
It's more likely there would be new challenges for the immune system, related to the completely new aggression for which it has no experience at all.
Just imagine what happened to people in America and Australia when first met illness brought by western invaders.
Climate will also influence how the organism will react: i.e. if one is weakened by prolonged exposure to cold, it's easier to get lung related sickness.
This for illness induced by an external agent. For those related to body ageing or malfunctioning (think auto-immune diseases or simply growing old) I think it is more likely that a new planet and its climate will at best do nothing, at worst it will kill us faster.
My reasoning for this is that our body has been optimized for what we have on Earth (gravity, pressure, light, temperature, oxygen, etc.) Any change to this will have effects on the short and on the long term. On the short term we know something from some studies (like oxygen toxicity or gravity deprivation), on the long term we can only speculate.
An important distinction is internal diseases vs external diseases.
Internal and external are not scientific terms for diseases, so lets clarify. By using these terms, I am alluding to Genetic vs Infectious Disease. But lets define our own terms.
Internal are those diseases that come from our own biology; diseases a person is predisposed to in their genes, or that come about as part of normal human life (examples, autoimmune, diabetes). External, are diseases caused by bacteria or viruses, (example pneumonia, HIV).
The same internal diseases like cancer, Parkinson's, etc will happen on other planets, and possibly at the same rates, but new internal diseases will be rare or non existent. This is because they are not related to the microbes around us but instead to our own bodies. If we are the same humans, then the frequencies of these diseases may change, but the types of diseases encountered probably wont. The obvious example of one that might change is more cancer with more radiation.
Externally caused diseases can be either bacterial or viral.
Diseases from bacteria are the most likely to differ from those on Earth. How they will differ will be determined by what kind of bacteria can grow in the new environments, and how their evolution may change because of this. For example if there is more material to decay on a new planet, then we may see speciation of bacteria that cause decay. This might lead to many more kinds of infections, and decay related diseases.
Viruses on the other hand probably won't change much. They are closely related to our own biology. So an alien planet likely wont have anything new that can affect us. There will only be what we brought with us, and any new strains that develop from them. How different these strains are from earth will be determined by changes in the way we deal with them on the different planets. If our doctors treat them in the same way on every planet, then they will stay the same.
Mark's answer is great because it classifies the many causes of diseases. Let me make a note on a group of diseases I'll call "parasitic diseases", extending Mark's classification of bacterial and viral to include all kinds of live parasites, because almost all biological kingdoms causes diseases (except maybe archae and plantae, but I wouldn't bet on it). There are diseases caused by fungus (i.e. candidiasis), by protozoan (i.e. brain-eating amoeba) and even by animals (i.e. cysticercosis).
Diseases caused by living beings vary greatly geographically (there was no flu in South America before the first contact with Europeans) mostly due to evolution. In isolated environments capable of supporting life, like spaceships and potential off world colonies, new diseases will certainly evolve, both as variations of preexisting diseases and, eventually, completely new diseases may evolve from previously unharmful microorganisms.