Rope(nylon) halters, can be ordered online, or some feed stores carry them
Leather show halters
Shampoo(dish soap works)
Brushes for washing
Spray bottles, filled with products(Kleen Sheen, Zoom Bloom, etc)
We use vinegar and conditioner mixed with water for a rinse after we wash them.
A few bigger things:
www.sullivansupply.com has most the supplies you will need. You could also try Craig's List. Talking to your local extension agent or someone experienced with showing can help get you started.
Those are just things I can think of right now, there's more.
If you have any specific questions, I'm sure everyone on here will be happy to answer them. It will definitely be helpful if you could find an older 4-H member who has experience and watch them, then try things yourself. You can probably get some information from the 4-H office, or your breed association. The Angus Assocation has a lot of information on showing beef cattle. The best way to learn is by doing it yourself, so don't be afraid to experiment. Go to a show or sale and watch them clip the animals. Sullivan Supply and Nasco are the best places to get products, I have found. Don't be afraid to start out small, even if you don't have the big fancy chute and trailer, you'll still have fun. Basic supplies you will need for the fair are feed and water tubs, feed scoop, feed, a hose, washbucket, soap (I use Wisk or Joy), scrub brush, rice root brush, scotch comb, rope halter, safety tie, show halter, show stick, show clothes (usually white or white top with dark pants or western shirt with dark pants-check the show rules), your animals registration and lease papers (if applicable), any other 4-H forms required by your county, a combable adhesive (like sullivan's EZY Comb), Zoom Bloom (Sullivan's). You can get a lot more in depth with the sprays and stuff, but when you first start you may want to not go all out. A blower is necessary, but they are expensive and if you can't or don't want to buy one right yet, see if you can share or borrow one. A chute is helpful, but not entirely necessary if you have a clean area to work where the animal can be restrained. Clippers will be important, but you can borrow those too. If your animal has dandruff you can blow in some Russ' Rag Oil a few times a week to treat it, or use a medicated shampoo, or use a rinse of one part apple cider vinegar, three parts water when washing. Good luck and I would be happy to answer any more questions you have.
yes, adults can show cattle. There are many open shows. At the fairs in my state at least, there is one day that they do 4-H or youth shows, and then the open show on another day, where anyone can compete.