First, there are lots of 'Thymes'. But the most common culinary thyme is Thymus vulgaris. It's a small perennial woody shrub or subshrub (having only some woody stems, near the base), native to the Mediterranean region. Most cultivars are quite hardy and easy to grow. They prefer dry conditions, but will tolerate areas with a bit more moisture if given good drainage. Even though perennial, it is not often long-lived in garden situations, but it is very easy to propagate. Layering, or simply burying the branches, and allowing them to root, works very well and you can keep thyme going like this for a very long time (no pun intended). You can also revitalize a bed, by removing, dividing, and replanting it. Thyme requires very little fertilizer and is not that picky about soils, though it does prefer looser well-drained areas. It's not particularly adaptable to indoor conditions as it grows best in high-light cool areas in the winter (imagine the Mediterranean in the winter). But you can put in inside if you like. Don't over-water, and keep it in the brightest area you can find - and if you can keep it cool, perhaps in an unheated room with south or west-facing windows, you will be most successful. However dried thyme is actually just as good as fresh, contrary to what some dilettante foodies would have you believe. Even fresh thyme is rather dry, and leaving it dry before using it actually activates some processes that enhance the flavor (even more true of some types of oregano, but also true for rosemary, and savory). I think some people hear how fresh herbs have much more flavor and adapt this to every species. It's so true with basil, cilantro, parsley, etc. that I don't often even bother with dried (though dried basil does have it's own flavor...). Now, I do use fresh thyme, for it's greener, sort of hay-like aroma, which dried lacks. Though the 'green' thyme is less intense and complex than fresh-dried thyme. You can't keep thyme forever once dried, however, and putting it in a well sealed container out of the light will help it stay flavorful. Don't grind it either, until use, as this will also hasten it's demise.
Oh, and you can harvest thyme, even in the winter, if you can reach it through the snow (you could put a cover over it...). It will come back vigorously if you don't pull up the roots.
