A travel health kit is recommended.
Take a copy of your prescription with you. This could be useful if you need to get further supplies, or as evidence that the medication is for your use. If your medical situation is even slightly complicated or unusual, then ask your regular healthcare provider to write a letter that explains it and lists your formal diagnosis. This letter can be useful both in explaining why you have these medications, and also to make it easier to replace your medications if they are lost, stolen, or destroyed at any point during your trip.
Keep your medication in the original boxes or bottles, complete with any pharmacy labels with your name on them. It may also be useful to bring receipts to prove when and where you purchased the drugs.
If possible, take somewhat more than you anticipate needing, in case your trip is unexpectedly extended. Theft of drugs from checked luggage is relatively common. On a short trip, keep all of your medications in your hand luggage. On a longer trip, when you need to carry a larger supply, consider packing some medications in your checked luggage.
The drugs that are readily available vary around the world. Ask your doctor or pharmacy to tell you the generic name (also known as the International Nonproprietary Name or INN) of any medication. Your doctor might also suggest an alternative that is more readily available in your destination.
If you need to get more medication when away from home, you may need to get a prescription written by a local doctor. You will probably have to pay for seeing the doctor in addition to the charges for getting the prescription dispensed at a pharmacy. This may be an unpleasant surprise if your medications are normally covered by insurance or a state scheme. Travel insurance is unlikely to pay out if you just run out, but may pay if your medication is stolen with your luggage.
Over the counter
Not all drugs that are available over-the-counter in your home country are necessarily available in your destination country, or they may only be available on prescription. If a need arises, you may need to find something slightly different to the one you're used to. Be sure to read all labels thoroughly so you understand what a medication does and how to take it. If you have medication allergies, be aware that some ingredients have different names in different countries, so check online for nomenclature worldwide. Get translation help if the label is not in a language you understand.
If you are in a country with very loose pharmaceutical regulations, there may be some over-the-counter medications that are not safe to take. Check online to see if a given medication is approved for use in a country with a strong regulatory regime (like the US, Europe, Japan, or Australia). In some countries, counterfeit medication is rampant, so avoid drugs with suspicious packaging and be aware of the situation in your area.
Be wary of taking antibiotics without a prescription, which can promote antibiotic resistance. Avoid items labeled "homeopathic", which are available even in many developed countries but which are not effective. Also beware of dietary supplements (which usually don't need to prove effectiveness and are untested for purity), especially those marketed for sexual dysfunction, weight loss, and muscle building.
Branded over-the-counter medicines can vary widely between countries. Don't assume that Brand A in a foreign pharmacy has the same ingredients as Brand A back home. Take time to read the box and check. Athletes have failed drug tests after taking an over-the-counter medicine which would have been permissible back home.
Some countries have harsh restrictions on medications, especially analgesics. Border controls might have harsher limits than the country itself. Opiates and opioids, such as morphine-based preparations, are usual illegal to carry without a proper prescription. In other cases, what's sold over the counter at home is something that requires a prescription or is completely prohibited at your destination. Check with the foreign embassy to be certain that your medications are okay.
Syringes and other equipment used with narcotics might be restricted.