Allergies and other kinds of substance intolerance are a problem for many people, and can be a real hassle while traveling. However, with proper preparation, one can still enjoy traveling safely.
Most airlines will accommodate your dietary restrictions when you book your flight. The choices will vary depending on the airline but typically include vegetarian, religious, dietary and child meals. Typically these meals get distributed first, before the 'main' options.
Alternatively, you can pack some food with you for the first leg of your trip so you can eat safely on the airplane. If your destination country doesn't allow you to import food, you can always throw it away as soon as you arrive at the destination airport.
Being cooped up in an airplane can expose you to allergens while airborne. For example, microscopic nut particles enter the air of an enclosed cabin whenever anyone eats them, and pet dander is carried into planes on other passengers' clothing even if no animals are in the cabin. There are few ways to mitigate this, although upgrading to a more expensive cabin class (with less dense seating) can reduce the amount of potential exposure. Some people with allergies pre-medicate with an anti-allergy drug (such as diphenhydramine, which is often sold under the brand name of Benadryl) to reduce the severity of any problems they might encounter during the flight.
A frequently used option for people with allergies is to stay in an accommodation which gives full access to a kitchen. This allows travellers to go to local supermarkets and buy what they need in order to be healthy.
On most trips, toiletries and clothes are easy to manage: Bring everything you need from home, so you don't have to use anything that you're uncertain of. If you'll need to wash laundry in another country, consider bringing along your own low-allergen laundry detergent. People with skin sensitivities or contact allergies can pack their own bed sheets and pillowcases as well, so that they can have a layer of "safe" fabric between themselves and the hotel bedding.
If you have potentially severe allergies, you should always carry medical alert information. If you're traveling where people don't speak your native language, then you need that information in the language of your destination. If you're not fluent in the local language, then get your needs in writing, on a card or piece of paper that you can hand to restaurant employees. Don't just write "wheat allergy" or "gluten-free" and leave it at that. If you have a wheat allergy, then write out that you cannot eat anything containing any form of wheat, including ingredients that contain some wheat, such as soy sauce and some sausages.
Always bring your allergy medicine, even if your allergies are seasonal, and it's the wrong season. Do check that your allergy medications are legal at your destination, or that they don't require extra paperwork. For example, diphenhydramine is regulated as a prescription-only medication in Zambia, and Japan permits it only in 10 mg pills (the 25 mg size is more common in much of the world, including the US).