In my experience, there are a few tricks to writing a smart character, for at least some values of 'smart', as 'smart' covers a lot of different traits that are often independent.
Well-educated: They will tend to use slang less and precise language more. They will not use larger words to show off (unless insecure), but they will use them to be more exact about their meaning. They will also tend to be either very knowledgeable about a field of study, or broadly knowledgable about many subjects, but few of them in depth.
Good at [Thing]: Depends on how extreme you want it to be, but the most common one would probably be math. How you would demonstrate this would probably depend on the story. If you have a lot of travel involved, and the group has traveled from A to B, and from B to C, but never directly between A and C, the math person could easily calculate an estimate between A and C (geometry/Pythagorean's theorem), noting "this does not account for terrain, so consider it a minimum distance/time."
Another story might find the math person figuring out the volume of a container with barely enough information ("hey, could you stand next to the tank a moment? Thanks... okay, that tank has about [value] gallons of (precious/dangerous liquid) in it.") Really, anything that might have you googling "how do I find out [value]?" and using your calculator app could do here. The character who is extremely good at math has memorized the formulas because they use them a lot, and can do them in their head, including long division.
Oh, long division in their head might be really useful.
Character A: Alright, so we have this much food and this many soldiers, so, um...
Character B: "X weeks and Y days of rations, rounded down, given a ration of [size] per day, probably less after accounting for spoilage. We can stretch it more if we use [smaller size], giving us X2 weeks and Y2 days, but spoilage will be even more of an issue so well-preserved food should be eaten last."
Bookwyrm Bookworm: Similar to well-educated, but even more widely spread and with spottier knowledge. So they'll know one or two very specific things about a random culture, and won't remember anything else they've read about that culture. Replace "culture" with any other subject as needed.
Fast Thinker: This one can be trickier to write, but is hypothetically really easy to demonstrate. You have some fast-paced event happen (such as combat or natural disaster) that has a lot of moving parts, and show the character first easily keeping track of the chaos and then passing on information/suggestions/orders to efficiently organize their group and maximize their effectiveness.
Combinations: This is where you run into the potential for ridiculous abilities of thinking, but if you have something like a very old Dungeon Core, I would expect them to be able to effectively drop into 'bullet time' mode where they are able to react to a lot of things all at once while calculating trajectories of arrows, analyzing the estimated power of a mage based on the effects of a well-known spell they cast, pulling up obscure information to use in the current situation, etc.