What is the best way to carry weights while walking to increase strength and endurance?”
If your goal is to carry small amounts of weight, then ankle weights or carrying them in your hands will work. If you want to get into greater weights then I suggest a well padded backpack with a good quality chest strap to carry some of the load. I’d suggest using a backpack with a built in hydration system, Blackhawk and Camelbak make several sizes and versions. This way you can carry a few litres of water as part of your weight and have water with you to drink along the way. The remainder of the weight should be as soft and as well distributed as possible. Don’t just throw your regular hand weights in the backpack and expect not to have a sore back at the end.