Because beach and lawn chairs will be primarily used outdoors, they should be made of weatherproof, very durable materials so they will last a long time. It would also be preferable if you didn't have to worry about bringing them inside or covering them every single time it rained!
You'll need to select the perfect material for the frame as well as a good material for the cushioning or webbing. Let's start with the frame:
A good frame should be made of aluminum, stainless steel, iron, plastic, or wood. If you choose to go with metal, make sure that it's properly treated so that it doesn't rust after a rainfall. Wood needs to be sealed, as well, to make sure that it doesn't rot after years of exposure to moisture.
Plastic may not be seen as very fancy, but it's actually the perfect low-cost, low-maintenance material for a beach or lawn chair. It's lightweight, too, which means that you'll be able to move the chair to catch the sun or to entertain in the way that you choose.
For cushioning or webbing, your options are likely going to be more limited: You're going to find options made of treated duck canvas or vinyl material. Both come in many different patterns and colors. Vinyl is more durable, but treated duck canvas is softer. A good middle ground might be to choose webbing or upholstery made from vinyl, and to have extra cushions or pillows in duck canvas for comfort!