Equipment Recommendations
—Buy ASAP—
Hand wraps
My recommendation are these:
https://www.titleboxing.com/protection/handwraps-speedwraps/title-semi-elastic-180-mexican-hand-wraps-1-pair-34844
Really any hand wraps will do, just make sure you get at least 180".
For a guide on how to wrap your hands:
https://www.youtube.com/watch?v=-kgVnNAHLLM
Gloves: Essential ASAP
Value pick:
https://sanabulsports.com/collections/striking-equipment/products/gel-boxing-gloves-essential?variant=34128789006
Sanabul has good quality gloves for the price.
$$$ pick:
Rival RS11V/RB10, Winning, & Cleto Reyes gloves are seen as the best money can buy.
Gloves with velcro straps are easier to get on and off than those with laces. For the club setting, velcro is preferred.
—Optional for Sparring—
Mouthguard
You'll need a mouthguard that is fitted like the one linked here. If you want to use a mouthguard you have from a previous sport, make sure it is fitted for both your top and bottom teeth. https://www.amazon.com/Venum-Challenger-Mouthguard-Red-Devil/dp/B00F2U1CTQ
Headgear
If you're new to the club you can wait a while before getting your own headgear. It will only be necessary if you eventually decide to try sparring (which I highly recommend for everyone at some point, but we'll start slowly with drills first).
Here is a budget option:
https://www.amazon.com/MaxxMMA-Headgear-Training-Kickboxing-Taekwondo/dp/B01356307A/ref=sr_1_2_sspa?dchild=1&keywords=boxing+headgear&qid=1631633708&sr=8-2-spons&psc=1&smid=A1HMSA0TDVNY1U&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUEyNEVLMjk1UTNVOUE1JmVuY3J5cHRlZElkPUEwMzgxMDc3MVRaMTgxS1A3VTdLNiZlbmNyeXB0ZWRBZElkPUEwMDQwOTE1MldEUFdYTk83NkhaUSZ3aWRnZXROYW1lPXNwX2F0ZiZhY3Rpb249Y2xpY2tSZWRpcmVjdCZkb05vdExvZ0NsaWNrPXRydWU=
More expensive options include the Winning FG 2900, Uma R9, and Cleto-Reyes headgear.