I have 1602 and 1503 both. I was actually just talking this morning about going to pick up 1701.
A basic rundown from 1602 and 1503:
Usually, you usually start with a boat with 50 tons of tools, wood, and food. The you found an island. Some islands have iron ore or gold on them in the mountains (Salt and Marble also on 1503) and each island can grow up to 3 different resources such as tobacco, wine, and sugar, etc. The resources grow based on the island type. Northern climate: Tobacco, Sugar, Wine**. Southern Climates: Spices, Cotton, Cocoa**)
** 1503 Changed some of these and removed Cocoa but added in a few more.
You also have basic resources like wood, furs, bricks, tools, etc. 1503 has a little bit more complex of economy and a better military system than 1602 did. It is mostly an economic/builder game. If you enjoy mostly playing as a builder in Civ then you would probably enjoy 1503 and 1701, perhaps 1602 as well but 1602 is better if you have not played 1503 and I am guessing 1703 will have the same effect on the previous versions. 1503 had a great campaign system.
The point is to build an island and begin to build housing for your inhabitants. They start at the pioneer level and require low level resources. As you supply them with more recreational buildings and resources they will go up to settlers, then citizens, then merchants etc. All the way up to aristocrats.
You will gain more options of things to build as they go up in level. Designing the layout of your island is where the real strategy comes in and it mostly is a laid back game I enjoy because I like the planning. The soundtracks are nice to for a day you want to relax on.
Pirates come into play and are usually the most annnoying enemy but you can also bribe them to attack someone else.
Military is pretty basic in 1602 and the AI is not very bright. I can wipe ut an entire island with only 1 cannon if I want, but I usually use about 5-10 to speed it up alot more.

1503 was improved upon by alot and I actually have to mix up my units. Also advancing on an enemy on the same island as you can be difficult and time consuming.
Your Navy has been so far the most important aspect of the game though of course militarily. I think checking out any of these would be worth it if you enjoy the builder strategy of Civ and don't mind longer games.