Determining stem material

acrylic is hard, chippy. Like if you dropped it, it feels like it might shatter. Ebonite or vulcanite is like rubber, but when polished looks deceivingly hard. If you tap it against your teeth and it almost hurts, then it's acrylic, and if it feels very comfortable to your teeth it is one of the rubbers.
Also, if it has color, like swirly bright colors, it's most likely acrylic. If it is black and shiny, it could be either one.

There are a couple popular tests you can use. If you flick the stem with your fingernail and it produces a sharp "tick" sound, that's acrylic. Vulcanite will produce a duller sound because of its softness. Another method is to rub the stem with your fingers until it gets a bit warm and then smell it. The smell of hot rubber indicates vulcanite.

Rub the stem gently a few times and smell it immediately after. If you smell sulpur/rubber...anything, really....it's vulcanite.

I think there are other plastics, softer ones, that are sometimes used for stems. These may not wear as well as acrylic, and most often show up on less expensive pipes. See how this one wears. It could be acrylic, or it could be some other sturdy plastic. Vulcanite has come to be a little bit of a luxury, although it does oxidize (turn gray or green and cloudy) over time, if not maintained or if left in direct sun. More expensive vulcanite usually oxidizes less, or more slowly.

If it is already showing tooth marks, I'd say it was a soft plastic. The stems of many Missouri Meerschaum corncob pipes are pretty soft, although it usually takes them a while to show tooth marks. The softie bits are useful and will keep soft plastic stems from chewing through quickly, but they do require some extra cleaning to keep from getting scuzzy. Some people make them from plastic tubing sliced in sections so they cost almost nothing, but even the ones from the pipe shop are often several for a buck. There are many good-smoking inexpensive pipes when you go shopping again. Look at Chapuis-Comoy pipes at tobaccopipes.com, one of our sponsors. These have good quality acrylic stems and the briar is above its price level. Rossi Vittoria pipes are sturdy good-smoking pipes at low prices. PC offers some of their Stanwells at extremely reasonable prices, and so on. Buy you a few softie bits. Learn to smoke your new pipe. You haven't made a mistake. There's always a learning curve. Welcome.

The quality--and likely the identity--of the plastic use certainly varies. My one briar pipe has a stem that is assuredly not vulcanite, but it's also probably not acrylic. Mine too easily acquired tooth scratches, though not nearly as quickly as the stems found on MM cobs. I purchased a couple rubber pipe bits to avoid further harm, and have found them to be quite comfortable since I tend to clench and walk with my pipe.

I've always used the "flick" test with acrylic vs vulcanite, or an examination of the button will sometimes indicate what you have after you've seen a few examples of each. There's some vulcanite/ebonite/whatever that is almost hard to tell, I have an Askwith that I swore was acrylic until Chris posted that it was a high grade ebonite. I wish I had known about the rubbing/smell test then.
For other materials on some American factory pipes that reacts to alcohol, some alcohol on the finger and then pressed against a wide area of the stem will indicate it is plastic (ABS maybe?). If the finger sticks to the stem it is plastic. I can't remember where I read that, but I've found it to be true.