In terms of time changes, an hour or two means that you may have to adjust eating times and go to bed a little earlier or later than you might usually, but your body typically adjusts completely in a day or two. I am about to face a 7 hour time difference and preparing my sleep schedule for that is going to be difficult. As soon as I got on my first flight this morning I set my watch to my destination time and am trying to eat my meals and sleep on the plane in accordance with the new time... but this may prove to be pretty difficult, especially because I don't sleep well on planes. What I will probably have to do tomorrow when I get to Joburg, is to not nap and use natural light to keep me awake and reset my body's internal clock, sleep when its normal sleeping hours in the new location. Hopefully the next day my body will be closer adjusted to what is normal there.
Eating at airports and on planes can be downright disgusting, if possible pack some food that is healthy and you know you like. One of my family's favorite things to bring are almonds, which help tide me over as a snack on a short flight or between meals on a longer one. At the airport it is possible to find something healthy, you just kind of have to search for it. Most of the sit down restaurants have leaner options but if you're in a hurry and want to take something on the plane then you can't rely on those. Among the fastfood restaurants really look for a place that has fresher ingredients, not ones with processed or greasy food. Sandwich places are probably your best bet because they make them right in front of you as opposed to being sent to the airport frozen then were reheated. Plus sandwiches are really easy to take onto the plane with you. Mexican food places are another somewhat healthy option if its a place with fresh ingredients like guacamole, pico de gallo, corn salsa, beans, lean meats, etc. Just don't get things smothered in queso or sour cream. Avoid any foods that may not sit well with you on a plane with a tiny little shared bathroom with people all around... Avoid fried food and overly greasy food like pizza or 'chinese' food.
Drink plenty of water. If you are trying to sleep on the plane then avoid caffeine but especially avoid soft drinks and carbonated beverages. I know you can't bring your own water through security but you can bring your own empty water bottle and fill it up at the water fountain conveniently located near the bathrooms. Having your own water means you don't have to rely on the tiny beverage they serve you from the beverage cart on the plane at certain times. Water will help you flush out toxins that you could accumulate on your travels and a lack of caffeine will help you adjust to time differences.
And now to working out. Most hotels have a limited gym with a least some free weights and cardio machines so you can always try to utilize that if you have limited time. One of my favorite ways to get to know a new place (if its safe enough of course) is to just walk around, or if you want to really get some cardio in then go for a run around the new city. Lots of websites will give you running routes for different cities, as will most hotels. If on vacation then you can usually schedule your own time, so try to schedule in some athletic or at least lively activities.
The key to traveling and staying healthy is to try to keep as many of the same healthy habits as you have at home. Go out and travel and enjoy your summers!