First off I have no clothes you’d ever wear to a gym. I wear jeans and a t shirt pretty much daily (think Hank Hill). Second, I don’t get what you do there. I hated gym time in school (workout gym, not like throwing balls and running around gym, thats fun) and I don’t get what you do. Run on a treadmill and lift some weights? I feel like I could do all of that at home. Gym memberships are insanely expensive. Are home workouts actually effective? Does one even enjoy gym time?
Captain obvious sneaking in, but There is tons of legit way to exercise, not just lifting weight in the gym, you can look for Volley-ball, Rock-dancing, or karate-class all of that is a legit way to exercise, which may even be cheaper and more fun than going to a gym.
Regarding your other points
Get a T-shirt and either a short or a pair of legging, to start-with you don’t need any of these high-end expensive fabric (Which may help if you plan to climb an alpine summit or run a marathon, but not to lift some weight casually). Total cost should be in the 10-15 EUR range (It sucks to live in a world where clothing is built by slave-like-labour).
Most gym would include a couple of coached session as part of their beginner package. But a typical program would be like warm-up on a threadmill/eliptical then a bunch of strength exercise either on machine or free weight, and 10 minutes of cool-down with stretching, you can modulate duration/intensity depending on your goals.
You can do a lot at home with body weight. However, being in the Gym sometimes helps as you are in a dedicated place with a dedicated time-slot. ideally choose a place either near your home, or on your commute, if not you’ll never go there