There are many kinds of rice. "Plain" long grain is the most common in the U.S.; there are a number of good packaged brands (Canilla, distributed by Goya, and Alma, a Greek brand, are good and easy to find in the Northeast) and it can also be bought in bulk in many stores. I always stay away from "converted" or "parboiled" rice, but I'll lump these with the above and call them "plain".
"Jasmine" rice is popular in Thai cooking; the Haitians seem to prefer it too, and Indian stores and many ordinary supermarkets often carry it in large bags (avoid the designer packages at outrageous prices). "Basmati" rice is an old popular type in Indian cooking; it is thin-grained and not sticky. These rices are both very aromatic (the word "nutty" is frequently used to describe them).
"Glutinous" rice is very sticky; it is used in Chinese cooking in many interesting ways, and in Japanese cooking commonly with sushi. "Arborio" rice is also rather short-grained and glutinous; it is used for risotto. I will not deal with these short-grained glutinous rices here, as I think of them as always being prepared with onions, or with seaweed, or vinegar, or other things that limit their usefulness as leftovers to be transformed into other dishes. Nor will I deal with unhulled rice, nor with wild rice, which is not rice at all. This is not in any way meant to be a value judgment.
I just want to give a few suggestions on what to do with (a) fairly "plain" rice, as I described above, and (b) highly "aromatic" rice, like Jasmine and Basmati, that have been simply boiled or steamed, and have left you with rather plain leftovers that would lose something if just reheated as is. The suggestions will be somewhat sketchy, for experienced cooks.
Bring to a boil, stir, and keep at a slow boil, stirring from time to time with a wooden spoon. Keep adding hot milk and keep at a slow boil, stirring from time to time, until it has become thick and the rice has lost its individual graininess and is rather soft. You will have to add a lot of milk, do not be concerned, and quite a bit more sugar than you might think is prudent. The exact amounts will vary, depending among other things on the rice.
The final result should be very runny when it is still hot, not too thick, and it will have a texture similar to tapioca pudding when it is chilled. Pour into a bowl or in individual serving bowls (the size of custard or berry bowls), let cool a little, sprinkle powdered cinnamon evenly on top, but not too much. Put in refrigerator (uncovered is theoretically ok, but it could pick up or give off smells that you don't want, so you could cover it with plastic wrap).
It is hard to get the right sweetness or the right consistency, but once you've done it once, the second time will be perfect. Most people do not like "rice pudding" because the rice remains too al dente and the milk has not picked up rice flavor, and because too much other stuff has been added. I guarantee that most people will like this rice pudding. One last point: the rice must have been cooked in water first (and without oil, as was said above); if you start with raw rice and cook it in milk directly, the rice flavor will not develop as well.
As a final footnote, I recently experimented using Rice Dream instead of milk. When I first tasted Rice Dream, my skepticism about food substitutes notwithstanding, I thought it was a very interesting product, but couldn't figure out what to do with it. All it shares with milk is its off-white color, and its cooler hue makes the difference all the more striking; I would never think of pretending that it is milk. Rice pudding, however, seemed a likely use for it, but, now that I've tried it, I do not recommend it. It does not thicken the same way; it's less velvety than with milk. In addition, its strong brown rice impact is too monochromatic and in-your-face, and misses the interplay of flavors that make this traditional dessert enjoyable. I would still like to discover an imaginative way to use Rice Dream, and welcome your suggestions.
Start with cooked jasmine rice (plain or basmati ok, but less good, and, if you are starting with uncooked rice, cook the jasmine rice without oil). Melt a generous amount of butter in a skillet, about 2 Tb per cup of cooked rice, and sauté some chopped celery (not too much; maybe 1 rib is enough for 2 cups of rice) with anise seed and turmeric to taste (or 1 tsp each for 2 cups of cooked rice). Keep it simple; don't add other spices. Add the rice to the skillet and stir to coat well with the butter, cover, and warm through. Taste and salt to taste; don't forget the rice may have started out already salted. Add heavy cream, about 2 Tb per cup of rice (same amount as butter), and warm through again. Transfer to a deep serving dish, and cover the top generously with roasted sesame seeds. Serve hot with a dry meat or vegetable, like fritters or tandoori, not a saucy curry, as the rice is moist and sticky. Works well also in non-Indian meals, but only with strong, spicy dishes that will stand up to that much butter and all the sesame seeds.
The "fried rice" that everybody used to order in Chinese restaurants before they graduated to plain steamed rice, is a good way to use a bunch of leftovers. If you are starting out with steamed rice that was cooked without oils, the rice must be stir-fried in some oil first, especially before adding any liquids to the rice, or you'll end up with mush. Once this is done, or if you started with rice that had oil to begin with, then you can chop and add anything you want to the rice and stir fry in a wok. You may add soy sauce, bean paste, chili sauces, and dry them by sitr-frying. Use all kinds of leftover vegetables, especially green vegetables, and meats that were cooked plain. Here are some things to add that work very well, along with your leftovers:
roasted peanuts or cashews (dry roasted or pan roasted)
Chinese or Japanese pickled radishes and turnips
scrambled eggs (pre-cooked)
cooked or uncooked shrimp (even if you already have pork or chicken as another ingredient)
garlic and ginger root
sesame oil (admittedly a cheap way to make anything good)
spinach or other greens in small amounts, pre-cooked
coriander leaves (i.e. cilantro)
dried red chiles (seeded and previously charred in oil --this can be the first thing you do in the wok)
Cooked rice is used in Sicilian arancini and other difficult preparations that merit cooking the rice from scratch. It would be silly to think of them as uses for leftover rice.