Hi, everyone. I'm new to studio photography, and looking for some help with how to mount products that I'm shooting such as watches, cologne bottles, etc.. I think I've seen that photographers have used some kinds of extension arms? I see all these clamps, and grips, and I'm not sure what it all is, or what I do/don't need. Any help is greatly appreciated. Thanks.
Your question illustrates exactly why the person filling the key grip position is always listed prominently in the credits for all movies. It takes a lifetime of practical, hands-on experience to become really proficient in knowing how to mount "things" whether they be heavy lights, cameras, large or small products, scrims, etc. Such people typically have been expert do-it-yourself artists before they are even considered to be hired as even an assistant grip. This is because a lot of mounting problems are solved by one-of-a-kind, hand-made mounting rigs.
Fortunately, for small product photography such as you describe, the bar is much lower. For example, to suspend a ring, just use some very small diameter sewing thread supported from an overhead bar or rod. A single thread isn't very good because it will let the ring swing like a pendulum, as well as rotate. Two threads attached to two separated points on the overhead rod will stop sideways swinging and rotation, but will still allow it to pendulum in the perpendicular direction. Add a third string, attached below and you have constrained all motion of the ring. Hint: don't tie a knot on the object, just pass a loop of string through it - it will be much easier to remove in PS. If it's a heavier object, use monofilament fishing line.
You should also become familiar with adhesive putty (aka, "sticky putty"). You can buy it almost anywhere and it is incredibly useful for product photography. Just put a tiny dot of it under a ring is often all you need to do.
In addition to building up a kit of common, every-day items like sewing threads of different colors, sticky putty, adhesive tapes of various sorts, you should be building up a selection of general purpose clamps. Some you can find at the hardware store, but you should look into more specialized small clamps for photography. Familarize yourself with what is available by going to www.bhphotovideo.com and typing "clamp" in the search box. For example, I hardly go on any shoot without 2 or 3 of these in my bag: https://www.bhphotovideo.com/c/product/824360-REG/impact_cc_106r_super_clamp_with_ratchet.html
For $20 each, they are a God-send when improvising lighting on location.
This is a big field and there is lots to learn. Start small, and work your way up.