Paperwork
If you get all your needed visas before departure it may take many months and even then, most must be used within six months of issue. It may be better to pick them up as you go. Most countries have consulates near the borders; check your guidebook for specifics. Keep handy a dozen or so passport photos, which you may need for applications. Make sure your passport is valid for the duration of your journey.
Health
Your physical state is an important day to day issue, since travel naturally involves quite a lot of strain on the body. Pace yourself according to what your body can endure.
If you take regular medication, bring along a supply or the prescription to get refills along the way. Overseas availability is not a problem in most developed countries. A doctor's note stating that you are taking medication helps at border crossings.
You may need shots and/or malaria pills in many developing countries. Consult your doctor or local health department for advice. Your guidebook should also list what is needed, although they may not be up-to-date. While overseas, check a consulate for recommendations and updates on disease outbreaks. Some countries now have AIDS testing requirements, usually for long-term visitors.
If you are going, say, from Europe to Africa or Australia to Asia, wait and get your shots a month or so before you depart to ensure they are as up-to-date as possible. On the other hand, some shots, like hepatitis B, must be done over 6 months. Common sense and prevention will be your best allies against getting sick overseas.
Should you fall ill, get to a hospital if possible. American and other embassies and consulates carry lists of local, English-speaking doctors. The biggest cause of tourist emergency hospitalization in many countries I visited: people wrecking their rented scooters.
Travel insurance is worth looking into; there are many, many travel insurance websites. In addition, your credit card or student ID card may offer some basic coverage. There is also evacuation insurance such as Medex (http://www.medexassist.com/ppcindividual.cfm), which covers the cost of airlifting you out of harm's way but does not cover actual medical costs.
These figures are very general and will probably not be borne out by your own travels. Once you get into a travel rhythm, though, it gets easier to save.