When I first started as a trainer, I was lugging every piece of equipment imaginable deep into Central Park. From medicine balls to ankle weights, nothing was too heavy to bring to my dedicated clients. Then I got smarter.
The thing is that when you're training outdoors (or while traveling) you don't want to recreate the gym experience. You want an effective workout where you can enjoy your surroundings and not be fussed with lots of gym toys - especially if you're the one that has to pack and lug them around!
These three pieces are my tried and true travel buddies. They've been through five years of wear and tear in Central Park bootcamps, searing hot sands of Rio beaches and cliff tops in Europe. They're all easy to carry, easy to pack and have so many workout possibilities.
(Note: I have used all of the brands mentioned and am recommending them based upon my positive experience with them.)
Inflatable so it never takes up much room, this pick is one of the most versatile pieces of equipment you'll ever own. I especially use these with clients that need some modifications due to lower back or abdominal injuries. Great for lower body work, core work and even cardio. My preferred brand is Physio World.
Mini bands are great for rehabilitation exercises (e.g. if you've been to the physio) and for adding resistance to traditional bodyweight exercises without having to use weights. I use them mostly for lower body - especially for working into the glutes and thighs thus you may know the better as "booty bands" if you've trained with me. They are also tiny and easy to pack away; just make sure to not forget them! I used to lose them the time in the park. My preferred brand is Perform Better.
This might be an odd choice and of the three, they will take up the most room but it's still a lot less then a pair of sneakers. I use these for cardio and agility exercises. You can also rely on rocks, sticks, etc on-site of wherever you do your workout, but those aren't always a given. These are bright, easy and professional (for when you want to show the world how dedicated you really are to your fitness!) I usually pack 5 with me and space them out into a single line. You can do lateral jogs, high knees, walking planks, etc. for timed intervals or rounds. I don't have a specific brand for these as my pair is YEARS old and it's faded out but most will work well.