The first thing you should know is that EVERYTHING (from coffee to lemon in Coke) comes in hot (熱 "yeet") and cold/iced (凍 "doong") versions, so any drink order would usually precede with one of those two words. Remember that if you're ordering a drink with a meal though, usually the cold version will cost you a few extra dollars. HK people are also all about the efficiency, so names are usually shortened to three words. For example, lemon tea is 檸檬茶 ("ling moong cha") but is shortened to 熱檸茶 and 凍檸茶, respectively.
Tea and coffee
As you probably gathered from above, tea in Cantonese is 茶 ("cha"). If you want just plain tea, you don't have to order it; waiters will set some down when you're seated at the restaurant. But what if you want something fancier? We've already talked about adding lemon to tea, but the most popular drink in HK-style cafés is milk tea (奶茶 "nai cha"). Don't confuse this with tea with cream and sugar though, here they add a substantial amount of evaporated milk. Sugar is added at the table; if you're used to artificial sweeteners you will need to carry around your own.
Coffee in Cantonese is 咖啡 ("ga feh") and will usually come with milk or cream already added (unlike the west, they don't have creamers/milkers at the table). If you want black coffee (that you can add sugar to after), you can say 齋啡 ("zai feh"/plain coffee) or if you're feeling a little advanced, you can say 飛砂走奶 ("fei sa jau nai"), which means no sugar or milk.
You can mix milk tea and coffee together to make 鴛鴦 ("yuen yeung") for a real caffeine boost. If you don't like evaporated milk, you can order 茶走 ("cha jau") or 啡走 ("feh jau") for tea and coffee with condensed milk added, respectively. (The latter is awesome, btw, one of my former roommates, who is not Chinese but Jewish-Russian, used to make it.)
Adding lemon to everything
For some reason, people in HK like adding lemon to everything. This could be normal, like adding lemon to tea or to water to make lemonade (檸檬水, or 檸水 "ling sui" for short). My personal favourite is hot "lemonade" with honey (熱檸蜜 "yeet ling mut"), which I sometimes make for myself with a little ginger when I'm feeling under the weather.
A trusted HK cold remedy, on the other hand, is hot Coke with lemon (熱檸樂 "yeet ling lok", the 樂 comes from Coke's full name, 可口可樂), which I tried once when I was younger and didn't really care for. For an extra oomph, you can add ginger to that too, but I'm not 100% on whether we're in common beverage territory anymore.
Other common lemon-y drinks that I'll cover in my next entry, "Drinks that haven't quite made it out of Hong Kong", include Ribena with lemon (檸檬利賓納, or 檸賓 "ling bun" for short) and lemon coffee (檸啡 "ling feh").
As a former British colony, some British drink mixes have made it big in Hong Kong as well. Ribena can be considered one (it's sold as a syrup to be diluted into a drink), but the ones I'm thinking of are Horlicks (好立克 "ho laap haak", but people usually just call it "Horlick" in a Chinese accent to meet the two-of-three syllables requirement), a malted milk drink that's supposed to be served hot (though you can find it iced too) and Ovaltine (阿華田 "aw wah teen"), a malted chocolate drink. While it's usually not found in restaurants, I must admit I was more of a fan of Milo (美祿 "mei look")—essentially an Australian Ovaltine—myself.
If you don't like barley malt in your hot chocolate, you can order a 熱朱古力 ("yeet jue goo lick"). Like the West, "cold chocolate" is chocolate milk, which is 朱古力奶 ("jue goo lick nai"), but who orders milk at a restaurant?
I think this covers most of the drinks you can get at a typical HK-style café. Is there anything else you want to learn how to order?