Public transportation in Bangkok isn't that complicated, except for using buses, but since there is no article here on it I'll add one. It makes the most sense to list out options by type.
The internet will turn up lots, but there is one official site that lists options: www.transitbangkok.com.
Most of what is written here will have some detail covered in that reference, but it won't be 100% accurate, not frequently updated related to changes.
Airport access travel:
Most people take a taxi, and 300 to 350 baht would cover transit from Suvarnibhumi to most places in Bangkok (US $10).
The Airport Link rail line is one notable exception; it would cost less for most people to use that (something like 40-50 baht for the local line), and it's relatively convenient. The main drawback is that anyone not travelling to near a stop would need to take another taxi as well, and moving luggage could be a challenge if someone has a lot of it. That prior site will list some details, but it won't keep up with changes, like mentioning that they dropped the Express line (but the local stops don't take that much time to run the whole length anyway), or that they extended hours to after midnight.
You can hire a "limosine" from the airport, which usually isn't actually a limosine, typically an SUV or something such. These make sense if you travel with a family with a lot of luggage because otherwise one would have to take two taxis, which in a way might also still make sense. These would generally cost around 1500 baht per a trip, quite a bit given a taxi would be no more than 400. It's probably better to use an established stand to hire one and not just someone that offers, since it's hard to say what the latter form of provider would be like.
These are required by law to accept passengers to all destinations and to use a meter, although taxi drivers won't stick to that. 100 to 150 baht gets you pretty far in Bangkok, so this is the easiest and most common form of transit used, by locals or visitors. Drivers do pretty well with picking up directions for not knowing that much English, which isn't really a given in all cities. They will sometimes ask if you want to take the freeway or not, and since the meter runs if the car stops moving it often won't work out all that differently. You might save a little expense for not taking a toll road, and spend a little extra time, but for a lot of tourists the extra 40 baht or so ($1 ) doesn't make much difference. Drivers should ask this to give an option but they won't always.
BTS (skytrain) / MRT (subway)
The other main forms of public transportation, easy to use, frequent, and inexpensive (from 15 to 60 baht depending on length of trip, with various alternatives for passes for multiple use at the offices at stations). The only downside is that the systems aren't as extensive as at some cities, like Singapore or Seoul, so it only works well if the origins and destinations are near stations. In some cases it could make sense to do a long trip on one, or both (they sort of link up in a few places, just expect to walk a bit to make that happen), then take a taxi or something else, because cutting off traffic jam delays is a major advantage.
Oddly these usually cost as much or more to take than a taxi, with rates that are always negotiated, there are no meters. Going a kilometer away, so close that you could walk, starts out around 40 baht, and from there it just depends on the driver. It may be cheaper to rent a tuk-tuk for the day, there is just a common theme that they'll take you to jewelry stores where they get a commission, whether you want to go there or not. Just my opinion but really everyone should take a tuk-tuk at least once, to have the experience, and although they are probably a little less safe than a car (it's like a golf-cart) you won't see bodies everywhere from getting thrown out of them. It does happen, though, so if someone is really drunk in one they should still keep their head about them, there are no doors.
I don't take these so I'm not the best person to fill in details, but roughly the same deal as tuk-tuks, but on a motorcycle. Because they work in groups, sort of self-organised, there might be more in the way of conventional pricing that emerges but probably nothing like set rates.
Ferry / Express Boat / Tourist Boat
A really cool alternative, since this gets you way out of traffic, and a different look at the city, and these are inexpensive. The Express Boat Line (one of 4 or 5), marked by an orange flag, only costs 15 baht (US 50 cents) and stops at most main piers, running every 15 minutes or so up until 7 or 7:30 PM. The obvious down-side is that these only work if you are near the river, or going to near the river. Some people complain of crowding on these in review sites, and if your luck or timing is bad the boat will be crowded, as will occur on a bus or a train. The Tourist Boat version is about the same, costing 40 baht instead of 15 for a boat not as likely to crowd up, with better announcement of the stops in English, with potential for both to get some sort of day-pass (I've heard), but at those rates it doesn't seem to make much difference or sense.
These could feel a little uncertain if someone is accustomed to being on the BTS (skytrain), where the little map and indicator lights make the trip stops very obvious, especially if they get planning completely wrong and want to go to a stop the boat doesn't make. Easy steps around that hurdle: take a picture of the map on the pier, then you'll have it on your phone, or get a SIM with data access and then Google Maps will keep telling you where you are, and of course it knows where the Express Boat piers are, even without searching "pier" to get these highlighted by red dots.
Khlong Boat (canal boat)
I've never taken one, but that main site lists out the map and other details. I'd love to, they just don't seem to go where I go. As with the river ferry it's not different than taking a bus or subway, just the boat version, but in general Thais would avoid this travel mode because they are concerned about drowning, or the water splashing on them, or just the smell (the water I really wouldn't swim in).
Challenge mode! That website tries to make it possible for someone not already familiar with a busing system to use one of the half-dozen different bus types, but it's not easy. If you could read and speak fluent Thai the bus staff could help but otherwise you'd really need to know what to take where, and even then checking on random buses with staff wouldn't be practical. Buses are inexpensive, with non-aircon rates starting around 8 baht, and aircon around 11, up to 30 or so depending on length of trip and the bus line type. There is no central bus system, no map or schedule of stops, no swipe card, so it's all pay as you go to an attendant.
Along the same lines as bus, really for commuters, from one central transit stop to another, all but impossible for a tourist to figure these out. So I'm really mentioning this more for completeness, that these do exist.
For around 2000 baht (not a fixed number, and that will keep going up) someone could hire a van and driver for a day, really not a bad deal given what a day's worth of taxi expenses would add up to. The catch is that this is a reference to services locals use, not tourists, so one would need to seek out a local service provider company. The limo / shuttle services mentioned in the airport section would probably offer the same but would surely charge a considerable sum for such a service, since a trip from the airport starts and not much under 2000 baht.
Uber / services like that
The wave of the future, right, the rest of this list will be completely replaced someday. They have such services in Bangkok, I just don't use them because taxis are everywhere, so they're not likely to save any cost or time at all. So why use one? These might provide marginally better rates for some types of trips (although a taxi will go really far based on rate charts that don't seem that expensive; you could take one up to Chiang Mai maybe). And they offer premium level services, so for people that don't want to be in a taxi that might work out better. As far as what app to use and other details, best to research that online.