At a minimum one should have proper riding boots and a helmet for riding. Beyond that I would say that you should have half chaps and maybe breeches for your lessons. I have some friends who only ride in jeans (even in lessons), but after I outgrew the "kid just having fun" phase of riding my instructor told me that I should be wearing breeches. She would have still given me lessons either way and I would have performed just fine in jeans, but it showed her a certain level of respect and presentation, and showed her that I was serious. Talk to your trainer about that. It may be nice to wear a nice looking shirt as well- doesn't need to be anything fancy, just something neat. You will be fine wearing either the tights or breeches, half chaps, helmet, gloves, and a neatish looking shirt!
Are you bringing your own horse or tacking up a lesson horse? If you are in charge of your equipment then I would clean your tack before your first lesson and use a clean and tidy saddle pad. You shouldn't necessarily be going for a show-quality appearance, but making an effort to look neat, clean, and presentable will go a long way in making a good first impression! It also never hurts to call and ask what is typical, and you could consider visiting the barn during lesson hours to see what other people are wearing. You don't need to be like everyone else, but you'll look off if everyone is wearing their best breeches with tall boots, a polo shirt, matching saddle pad, and spotless tack when you come in ratty jeans, a paint splattered shirt, slobbery pad with holes, and grungy tack. I'll be the first to ride looking sloppy, but you want to at least look neat and clean for the first time!