The Sunken Dragon expansion (comes with the Gold version) is the Campaign mode, anything else is sandbox or scenario play. I've played through most of Sunken Dragon and a few scenarios, but mostly sandbox mode. The campaign has a story to it, and a lot of interesting things going on that you wouldn't normally encounter in the sandbox mode, so it can be a nice diversion unto itself. Once you beat a stage, you can go back and retry it on a harder difficulty if you want to.
The sandbox game starts with whatever settings you give it - kind of like Civilization might - world size, island types, resource quantities, AI opponents, etc. If you choose to, you can start with just a ship (with enough goods to settle) and the map exposed (so you don't have to scout every island, which isn't difficult to do but is time consuming). Doing something like this can really let you choose a good strategic starting point - or you could let the game challenge you and place your first settlement for you, which might be less than ideal.
Every person in your settlement has needs - basic and expanded. Meeting their basic needs keeps them happy. Meeting their expanded needs lets them evolve into the next tier. As you gain more colonists, different building options open up - which lets you 'naturally' evolve your colony over time. You can't jump out of the gate and go for Tier IV units by hammering research to it - you must play the game and build a thriving colony to win.
You can set tax rates for colonists, which isn't about fiddling with percentages and dials - but rather selecting a dwelling type and sliding a bar. The happier a household is, the more you can tax them before they start resenting it. If they're elated they evolve, increase population, and generally have a good time. If they're only happy they stop evolving, but continue to increase in population. If they're merely content, they don't evolve or increase in population (this is the "yellow" zone, where you'll likely keep taxes while you're building to meet needs for the next evolutionary step, and then you'd decrease taxes to encourage evolution). If they're unhappy, they start leaving your colony. If they're angry, they'll leave in droves and even riot - destroying buildings as they do.
You start with basic buildings - pioneer huts, logging camps, fishing huts, game hunters, and a few other basics. As you increase your population, your available structures increases. Get enough Pioneers - whose basic needs are just food and a settlement, and you'll earn the ability to make Wool and Cloth, or place a Church. Once Pioneers have Cloth and Religion, they evolve into Settlers. Settlers have everything Pioneers had for needs - basic and expanded - as their basic needs; only now Settlers have additional expanded needs. Give them Tobacco, Schools, and Alcohol and you'll get Colonists. Colonists have all the needs of Settlers - Food, Settlement, Religion, Cloth, Schools, Tobacco, Alcohol - plus an additional set of expanded needs. It goes on like this through two more tiers - Merchants and Aristocrats; each needing everything the group before it did plus some extra. The final Aristocrat level produces houses with the highest populations and the best taxability - but their needs are equally large. Your colony will likely span across three to four islands with multiple automated trade routes running between them in order to support Aristocrats.
There are supply chains to build and design to use your island space optimally. For example: to produce Cloth efficiently, you need two sheep farms. The sheep farms could be on a different island with a trade ship hauling wool back for production - or you can place the weaver right next to the sheep farms and he'll walk over and collect the wool for himself. Building placement is a concern, but what's most important is building solid supply chains. If you've got an island full of Cattle Farms producing cattle, you could effectively ship them all to your mainland with enough ships and keep all of your Butchers going strong; if you lost a few ships or interrupted that chain somehow, your Butchers would faulter and your food supply would start shrinking.
The goods you produce get consumed regularly. Online guides can tell you exactly how many of what you'll need to support certain populations, but having an overstock means security and the ability to trade excess for cash, weapons, or needed supplies. As you increase in population, their demands will increase with them - you'll need to produce more food, more cloth, more everything. This means that your colony size is dictated mostly by your ability to control land and defend what you've got. It's also possible to trade with AI players to get what you need, but you shouldn't rely on this to keep your colony evolving.
On top of the economic simulation, there's an underlay of naval and land combat. You can produce two kinds of warships to defend or attack with, and a few kinds of troops that can be used to protect your islands or attack your foes. The combat side of the game is the 'weakest' - as it typically is in games with a focus on building - but it's functional. It's been my experience that both kinds of combat really just boil down to who has more cannons/spears/swords.
There's also an element of research - once you build schools they generate research based on the population around them. You can make Weavers more efficient, learn how to demolish rocks, uncover mining techniques that can unlock unlimited resources (a mine needs to be "unlimited" before this is researched for it to work), get better ships, and even learn a little about spying and stealing from your enemies.