Getting the gear that you'll be wearing is just part of what you'll need. Here's the rest.
Hockey Bag
I loved my hockey bag. It was a huge monstrosity, but when I lugged that thing around from rink to rink it made me feel like a player. My bag had two large side pockets at each end of the bag, and if I remember right, one little pocket for my wallet (though I usually kept my wallet in my pants, which I rolled up and tucked into the bag). I can't remember the logo on the bag, but it was black, and the logo was in a faded purple. I think I even had green straps.
What I've seen today from hockey players before the public ice skating that I attend are roller wheels on the bags, enabling players to roll their bags around like luggage at an airport. I suppose if I were playing today I'd grab me one of these newer bags. The older I get, the more I realize that anything that relieves back strain is a welcome change, and these newer bags certainly help with that.
Water Bottles
Bring two. I remember one game where my bottle fell to the bottom of the bench, and before I could pick it up, I stepped on it, slashing it open and spilling water everywhere. Not having water is "a bad thing." Get two.
One other consideration: decide if you want a straw, or if you want a plain nozzle. Me? I sometimes took my gloves off between shifts and unscrewed the top.
Tape/Velcro (for Shinguards)
You'll need some "regular" tape (different than hockey stick tape) for securing your pads to your body. I remember players using velcro to secure their shin guards, and I thought that was pretty economical, because you could reuse it. I never found any velcro for myself, but I didn't try very hard. Instead, I bought regular tape in bulk, because this gets used on a regular basis.
Extra Laces
I remember seeing a teammate break a lace on their ice skate, and he didn't have a spare. Of course this happened right before a game. In a fit of ingenuity, he tied both pieces back together in a basic knot, but the shortened lace prevented him from lacing up his boot as tightly as usual. Lesson: Keep at least one spare in your bag. Your teammates will thank you. (Make sure to find out what size laces your hockey skates accept! It's OK to have laces that are too long, but not the other way around.)
Sandals
After a game, some folks hit the shower. I rarely did. Adult hockey games are usually late in the evening, so I typically showered at home. But if you do hit the showers, you may want to wear sandals ("slides"). I've seen guys bring sandals, and I thought it was smart considering the floors in some of these rinks are not the cleanest I've seen.
Mouth Guard
I never used a mouth guard, but there were guys on our team that did. I figured, since I was wearing a cage helmet, I really didn't need to protect my mouth. I don't know if I was "lucky" that nothing serious happened to my mouth playing this sport. Consider your own health as you decide whether to wear one or not. As a beginner, the game may not be "fast", but it's still a contact sport!
Pucks
Keep a few of these in your bag. In pick-up ice, you won't have a bucket of pucks with which to warm up with. Having a puck or two allows you to contribute to the scrimmage.
Like golf balls you find in the woods, you can sometimes find a puck or two kicking around in the dark corners of your ice rink. If you see one, it's "finders keepers", but expect to lose your share of pucks during practice sessions and even games.