This is a tricky question.
Plain coffee should be gluten-free to low levels, assuming it hasn't been cross-contaminated by gluten (we'll consider flavored coffees in a minute). However, lots of people who follow the gluten-free diet report gastrointestinal symptoms from coffee.
So what's going on?
Well, coffee can be hard on your digestive system. Regular coffee contains caffeine, which can lead fairly quickly to a bad case of the runs, especially if you're newly diagnosed with celiac disease or gluten sensitivity and your system is still irritated.
In fact, some people with irritable bowel syndrome report that plain caffeinated coffee triggers IBS symptoms, and I've seen a few reports from some IBS sufferers who say even decaf coffee (which does contain a tiny amount of caffeine) causes problems. (Here's more on IBS and celiac: Is It Celiac Disease or IBS?)
If you're new to the gluten-free diet, you might not be able to tell the difference yet between your symptoms when you get glutened and other digestive maladies (such as a bad reaction to caffeine). You might want to consider cutting back on your coffee consumption for a while to see if that helps with your symptoms.
Yes, It Could Be Gluten in Your Coffee
Of course, it's also possible that you are, in fact, reacting to gluten in your coffee. Even plain coffee beans can be cross-contaminated if they're processed in a shared facility or on equipment that also processes gluten-containing ingredients.
When you add creamer products and sugar to the mix, your odds of a glutening can rise substantially (for example, only some Coffee Mate products are gluten-free see the details here). I also know of one person who got glutened by a sugar bowl that, unbeknownst to her, had been contaminated with a flour-coated spoon during baking.
If you're getting symptoms from your plain coffee (and especially if you're pretty certain those symptoms stem from gluten ingestion), rule out the creamer and cross-contamination from your sweeteners first.
At that point, if things haven't improved, you may need to switch coffee brands. You also may want to consider buying plain coffee beans and grinding them yourself ground coffee offers more of a chance for cross-contamination at the factory level, simply because it's more processed.
What About Flavored Coffee - Is That Gluten-Free?
Coffee beans or ground coffee that you buy pre-flavored (those yummy-sounding flavors like chocolate hazelnut and almond toffee crunch) are likely to be considered gluten-free, and may even be labeled "gluten-free."
But that's not the end of the story.
Coffee flavorings generally are made with a proprietary blend of "natural flavors." Despite a well-justified fear of that term on labels (since it can hide gluten-containing ingredients, most commonly barley-based flavorings), it appears we don't need to worry about "natural flavors" in this context those used in coffee are rarely, if ever, derived from gluten grains.
However, many coffee flavorings have an alcohol base ... and that alcohol typically is derived from grains, including gluten grains.
The conventional wisdom among some (but not all) celiac disease and gluten sensitivity experts is that distillation removes the gluten protein from the alcohol, and so alcohol is considered gluten-free even if it's derived from gluten grains. (Read more on this: Is Alcohol Gluten-Free?)
However, many people experience gluten reactions to distilled grains.
The amount of grain-based alcohol in flavored coffee is miniscule even if there was some residual gluten left in that alcohol, it would register way below the 20 parts per million that's generally considered "gluten-free." But a miniscule amount is all it takes for some of us to react (see: How Much Gluten Can Make Me Sick? for more information).
If you do fine with flavored coffees, that's great. But I'd exercise caution if you haven't tried them before, especially if you tend to react to gluten-based distilled alcoholic beverages or seem particularly sensitive to trace gluten. You also might consider making your own flavored coffees at home, using alcohol-free flavoring (I've done this myself with Singing Dog Alcohol-Free Vanilla Flavoring).
There's no reason you can't enjoy a decent cup or two of java (even flavored) while following the gluten-free diet, providing you take a few precautions (you can even go to Starbucks if you follow my Starbucks Gluten-Free Guide). Just pay attention to your body, and be prepared to make some changes to your coffee habit if that java doesn't seem to be agreeing with you.