As mentioned already, XCOM is divided into two modes - strategic decision-making that takes place inside your underground base, and tactical turn-based combat that plays out in isometric, grid-based, semi-random mission areas. The strategy mode is perhaps less involving, but just as critical. All of the actions you take need time to complete, and as you scan for UFO activity around the world, events will occur, ranging from completion of your research projects and construction, to requests for help from various nations, to all out alien invasion. Though it's deceptively simplistic on the surface, the cumulative effect of your decisions is very real and the consequences of them can often be felt hours in advance, making careful planning essential.
You primarily expand your operations by building new structures. Though not as involved as base management in some strategy games, once again the simplicity is deceptive. This is primarily due to the adjacency bonuses you receive for placing certain structures near each other, and the time and money it takes not only to build your base, but to dig the spaces needed to house them. You will need to constantly trade off whether you want to get something immediately, or wait a little while for a better benefit. The same applies to upgraded versions of existing facilities - while you can build a structure that can handle deployment of four satellites instead of one, waiting to get to the point you can actually build it might not be worthwhile.
Beyond building structures, most of the game's strategy is a balancing act of keeping your soldiers well-equipped and well-trained, improving your existing capabilities, and keeping the XCOM nations happy. This is always easier said than done, as you have very limited funds at your disposal. While early in the game time and money aren't very pressing concerns, after an hour or two you will find yourself having to answer questions like "which nation do I save, and which ones do I leave to the aliens?", or "should I risk sending in a squad of rookies while my better soldiers recover from their wounds, or sit the mission out?"
The nations under your wing will respond to your actions or inaction through increasing or decreasing panic levels. Spread your resources too thin, and you won't get the better bonuses that come from pleasing specific nations; focus too much on a single nation or continent, meanwhile, and others will abandon XCOM, taking their resources with them. Everything eventually comes down to keeping the world, as well as your mysterious superiors happy, and the way that every decision you make directly or indirectly ties into that is just as compelling as it was in the original game. Eventually, something's got to give, and making sacrifices, especially on the harder difficulty levels, is critical to winning - playing "perfectly" is basically an impossibility. In this respect, XCOM is one of few modern games where failure is not a result of obvious mistakes or inaction, but something unavoidable that must be dealt with effectively to succeed.
Once you have finished up with your base management for the moment, or have had a mission thrust upon you, you will find yourself deploying a squad of soldiers to fight aliens on the ground. Missions take a few basic forms. Alien abductions are simple "kill all the bad guys" affairs; bomb missions require you infiltrate an area, locate and shut down an explosive before the ticking clock runs out; VIP missions see you escorting a civilian from point A to B; and terror missions require you save as many civilians as possible from an alien attack. Mission variety is solid, as the terrain you will play on is built up of semi-random "chunks" stitched together seamlessly. This is one place where the remake genuinely improves on the old, as the varied objectives make you change up your play-style and enforce different constraints on you.