Former Starbucks employee here - be really careful with the new teas. Many - iced AND hot - are not compliant. Order a brewed coffee, standard iced coffee (request 'No Classic'), americano, or espresso. All of these are some form of just beans and water. Teas... ask to see an ingredient list.
Brewed coffee - available in 8-12-16-20oz sizes - blonde roast (rotating flavors) has more caffeine; Pike Place is very medium bodied and even from a flavor perspective; dark roast (also rotating) tends to be very bold and flavorful, much darker than others (especially Italian, French, and Espresso roasts - if you're a newbie to black coffee, you might want to avoid these).
Iced coffee - available in 12-16-24-30oz sizes - you must ask to have the drink without Classic, and even then, I'd give it a smell before drinking it to make sure it wasn't added by mistake; also note that some of the substitute milks, like almond and coconut milk, will not be cold-drink-friendly; I've seen them produce chunks in iced coffees, which aren't so easy to drink through a straw . This is my favorite drink, though it can be a little tart; this would be easier/more approachable for many of you than hot black coffee.
Clover brew - This is a type of coffee that is brewed by the cup. You can order it hot or iced, with sizes and coffee types available dependent on location. It's like the two above, but it's a MUCH smoother cup of coffee, and will make the other stuff taste like dirt. It's also more expensive but it's a nice treat if you're a coffee aficionado. If you see Kona available, just order it iced, don't ask questions. It's heavenly.
Americano - available both hot (8-12-16-20oz sizes) and iced (12-16-24oz sizes) and is simply just espresso shots, water, and ice (if you ordered it iced). Shot distribution is 1-2-3-4 in hot sizes and 2-3-4 in iced sizes (so a venti americano has four shots).This does not automatically come with added sugar or milk. This tastes like a smoother cup of coffee - less variation in the flavor.
Espresso - simply just espresso shots - you can order a solo, doppio (sounds like "dopey-oh"), triple, or quad; you can get them hot or over ice; it's basically a tiny, super-strong cup of coffee. You can also get as many shots as you want (I've watched people consume 10 at a time), but beware... these babies are 75mg of caffeine each. If you have more than four, and you're not used to it, you might experience side effects like the head-spinning seen in The Exorcist.
If you're a tea drinker at home, like me, these teas are mostly compliant, pretty delicious, and available in most local grocery stores I've seen (though you will need to read the label on these, as with anything else during your Whole30): http://www.republicoftea.com/(Ginger Peach Green Tea is my favorite)
Just as a note: I know the menu can be a little intimidating at Starbucks, and no one wants to be the one at the register asking lots of questions with a huge line behind them. I'd be more than happy to answer any questions you have about these, other drinks, Starbucks in general, how to talk to the barista, etc. My six-year stint with Starbucks covered everything from barista to store manager.
That tea isn't compliant. I think most of their flavored iced teas are not going to be ok, but a lot of their hot teas will be. You can always ask for a steeped hot tea and a large cup of ice and make your own iced tea