Less is more - pare down to the absolute necessities and that includes what you leave in your car and locker. I tended to be a combination lock person, but your idea of wearing a key is good. A sturdy lock is a must.
Some of the cardio machines will have a holder for your water bottle and some have a little compartment that your phone will fit in. If I was using a weight bench, I stuck my water bottle under the bench to keep it out of the way. Or if I was using DB's, I'd put my water bottle under the rack. Hopefully I got home before the cooties from the floor climbed to the top of my water bottle, LOL!
I have an ancient iPOD Nano that I'd take to the gym and I'd stick it in my waistband or waistband pocket if I had one. I was fortunate that my last gym was within walking distance, which helped reduce the amount of stuff I needed to schlep with me.
Historically I saw people wear their work-out clothes in the sauna, but these were all dry saunas and not infrared. I'm jealous that you have access to one of those!
The gym should offer at least a quickie orientation so feel free to run your questions by the staff as they may have some helpful tips specific to their facility. I hope it's a good experience for you.