Well, PPG (personal protective gear -- the lab coat and goggles, for example) should be provided by the lab, to be certain that it meets the safety standard the hazards at the lab would need. So don't bring your own. You should dress light, in layers, so a lighter shirt and a light sweater, so you can take a layer off depending on the temperature in the lab. Lab temperature is kinda an afterthought when labs are built, they can have too much air conditioning, or get too hot because of waste heat from instruments and computers. Be ready for this, so your first few days aren't uncomfortable for you, and you're distracted when you need to learn what to do the most.
Try to select clothing made from natural fabrics ... wool and cotton burn slower than synthetics. You probably won't be at the center of a major accident your first day, by try to develop a safety consciousness in advance. Wearing a fluffy acetate or rayon sweater is a very bad idea, all the more if you handle flammable reagents.
If you expect to work with strong sulfuric acid, avoid denim blue jeans. The short staple fibers are very prone to being destroyed, and a minor spill will leave a hole rapidly. Any other cotton fabric will resist it a little better, and there are also scotch-guarded fabrics if you really suspect spills might be a problem.
Try to wear good quality shoes. Sneakers aren't a very good idea if you can help it. What you want is shoes that won't rapidly wick in a spill, so you have time to react.