For a beginner, I think one pair of trainers is fine. A pair of gloves - gym gloves - are useful for the weights to avoid making your hands sore.
I always wash my top after going to the gym. Leggings depends a bit on what I’ve done. If it’s outdoor Bootcamp then yes they’re washed afterwards because they’ve picked up dirt from the ground. Two hour gym session, washed again. A quick half hour in the cross trainer, I’ll probably wear them again. (Which will no doubt horrify half of MN.)
You don’t want too much of a full stomach before a class so I usually eat afterwards. If I’m going after breakfast, I usually leave an hour after I’ve finished eating before going. But if I’m out orienteering (which will take me up to 90 minutes) I’ll eat something like an energy bar beforehand so I’ve got enough energy to keep me going.
I don’t sanitise hands between machines but I am wearing gym gloves when using weights. However I always carry a towel and wipe away sweat, my gym has paper towels and cleaning spray you can use too. Please wipe down after you’ve used a machine.
It is a bit weird to be a member of one gym and go to classes at another. The exception being a member of a private gym and using the council leisure centre for some others (eg I know someone who uses the private gym for weights but does yoga on a pay as you go basis at the council leisure centre).