My favorite times are the mid-to-late Middle Ages and the late Industrial Age. Overall, I like the Middle Ages better, but I like them all.
Ancient Age - I love exploring, setting off into the unknown, discovering things, tracing my growth. Also, Ancient Age warfare is hella fun, even if it does constrict your growth somewhat. Ancient Age music is also my favorite. It makes me feel like I'm really exploring the world. My least favorite part is trying to figure out exactly where to settle and connecting everything with roads and such.
Middle Ages - Towards the beginning of the Middle Ages, I'm usually a tiny, fledgling society that's losing money rapidly and struggling to keep the empire in line. However, it's not long before I begin to emerge from the dust with knights and musketmen. I finally get over my woes with awesome wonders (such as Adam Smith's Trading Company) and with improvements such as universities and banks, and by the end I'm dominating the world as I begin to introduce cavalry into my army...the music is also very good, but when I play it seems to repeat the same song over and over and over again. What I don't like about the Middle Ages is that before all of those great improvements I'm usually struggling, even if I am dominating my neighbors through military force. Towards the end of the Middle Ages, your workers also tend to just lay around with nothing to do.
Industrial Age - The Industrail Age is tedious in the beginning, when all my workers have to start up again and build railroads, and the music changes to soft, melodic Industrial Age piano tunes (which I am not a big fan of at all). With riflemen and infantry, combat breaks down into defensive wars, which are my least favorite kind. However, you also begin to discover the entire world, forge alliances with civilizations, and watch grand civilizations crumble into dust under the combined might of the rest of the world. The world becomes completely known to you, and you can keep track of every event that goes on in the world. Epic AI wars are commonplace, and are very fun to track. Towards the end of the age, tanks, bombers, battleships, carriers, marines, and all that good stuff comes into play and war becomes much more interesting. Interesting knew advances, wonders, and improvements are introduced as well. By the end of the age you're on your way to conquest again.
Modern Age - The music once again changes to great stuff, the catchy, upbeat techno tunes which makes it feel like you're advances and that quickens the pace considerably. Warfare is completely epic, with modern armor, motorized infantry, carriers, and bombers dominating the map. However, managing your empire can become a real pain, as production in your core cities reaches the point where producing military units and improvements are a waste of shields. Wonders are mostly useless and no one can be trusted (which is fun and not fun in an odd if totally sensible way). The pace picks up considerably and, eventually, you don't have to worry about building improvements or railroads anymore (for the most part). And, of course, there's nukes.
You can't forget good 'ol reliable nuclear warheads.
Overall the mid/late Middle Ages and Late Industrial Age are my favorite times, and the early/mid Industrial Age is my least favorite time. The Ancient Age is very close to my favorites, with the Modern Age not far behind.
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