Most common serious ailments that affect travellers are contracted either by the bite of an infected animal or insect, by ingestion of contaminated food or water or by close contact with infected individuals. The last of these three is very difficult to avoid so those at specific risk should be vaccinated where appropriate, but certain measures can be taken to avoid contracting an illness transmitted by the first two routes.
Ticks can cause several very serious diseases and generally live in low growing vegetation and scrub. If you think you may be at risk because of trekking, wear long trousers tucked into your socks and preferably apply a DEET based insect repellent to both. You must also check yourself regularly, at least once every evening. They tend to head for the groin before starting to feed and the longer they are there the deeper they become attached and therefore the harder they are to remove. They should be removed gently by holding behind the head with tweezers or fingers and rocking patiently until they release their grip. Be careful not to leave the head or mouthparts imbedded as this may lead to infection. Apply antiseptic and wash your hands. If a rash or fever develops seek medical attention.

A few tips for keeping healthy:
The last point mentioned here concerning drinking water is of vital importance and bottled water should be used where possible. That said, there is an increasing trade in 'fake' bottled water which may even have a suitable seal on it, so if there is any doubt drink fizzy water (too expensive to fake), chemically treat your water (see below) or boil it. Water brought to a rolling boil and kept there at whatever altitude for 15 minutes will be safe, but is expensive in terms of fuel and may not always be a viable option, therefore chemically treating your water using one of the following methods may be the best option.

Should you apply all of the above tips and still become ill, management of travellers diarrhoea is very important. The first thing to do is to maintain a good level of hydration by frequently drinking freshly prepared rehydration solutions, or if these are not available a suitable alternative can be made by dissolving eight level teaspoons of sugar and one level teaspoon of salt in a litre of drinking water. At least one glass after each loose stool should prevent any severe dehydration from occurring. The use of loperamide and other 'blockers' should not be a matter of course as they will only decrease the frequency but not necessarily the volume of your diarrhoea, and may even make it worse in the long run. They do however have an important role to play when travelling on a bus or train where there may be many, many people sharing one toilet and under these circumstances their use can be invaluable. Once you have reached your destination however, they should be discontinued.
Danger signs would include blood or mucus in the stool, fever or prolonged severe diarrhoea (more than 12 loose stools per day) and under these circumstances medical attention should be sought.
Antibiotics can sometimes be used to treat severe travellers diarrhoea, these may be available wherever you may be, or they can sometimes be prescribed by your GP or travel health consultant prior to departure providing you are confident under what circumstances they should be used.
Some people are more susceptible than others to becoming nauseous whilst travelling, sitting in the front seat of a car, or towards the front of a plane may help, likewise being in the middle of a ship. Never attempt to read whilst feeling motion sickness. There are many brands of 'travel sickness' tablets on the market and your pharmacist will be able to advise you on the ones most suitable for you. Alternatively ginger tablets or capsules seem to provide some relief, as do wrist-bands that work on acupuncture points. Both will be available through local pharmacies.
There are many tales around about how to prevent jet-lag, but very few have any proof attached to them. There is much talk of melatonin (a hormone released by the brain just before sleep) but it is currently unavailable in the UK. There are a few tips that may make the transition to a new time zone easier:

When first arriving in a tropical climate the high temperatures and humidity may well seem quite oppressive and you will sweat profusely, this can consequently lead to dehydration and lethargy. It will take about three weeks for you to adapt to your new environment, therefore during this settling in period it is advisable not to over exert yourself and maintain a high fluid intake of safe drinking water. Throughout your stay there are a number of other points that should help you stay healthy in the sun;
When travelling to altitudes above 3,000m it is imperative that the ascent is made slowly with time to acclimatise below this level. If this is not possible most individuals will experience breathlessness on even the most minor exertion, possibly with headache and nausea. Paracetamol should relieve the headache but if it is too severe and combined with nausea, medical attention should be sought. At least three days should be allowed for acclimatisation at this level before strenuous activity, but if this is not possible there is a drug called acetazolamide that may be help in some circumstances. It is available on prescription only and must therefore be discussed with your GP or travel health professional before you go. To limit the effects of altitude sickness:
Practice safe sex - take contraceptives with you and always use a condom. It's a joint responsibility so take control. Always look for the British kite-mark and make sure the condoms are in date. Aids is a real danger all over the world.
Before travel, it is likely that anyone would say that there is no chance of them having sex whilst travelling - its too risky. However, it is a fact that in a relaxed atmosphere after a few drinks the unexpected will often occur, and it is always better to be prepared than to run the risk of pregnancy and sexually transmitted diseases, not least of all HIV.
You may think that the partner you have chosen seems nice, a nice bloke / girl, but the last person that they slept with may not have been! Condoms will not be available in many countries throughout the world and those that are available may be of poor quality due to bad storage or old age.

And finally...
Take your own supply of hypodermic needles in case you need an injection or to have blood taken - there's a slight chance hospitals in some less developed countries might use needles which haven't been sterilised properly.