I have tried a million polenta recipes, but this one is the only one I can find that is foolproof.

http://www.nytimes.com/2009/06/08/health/nutrition/08recipehealth.html?ref=polenta

It works for soft polenta if you serve immediately, and if you make it in a small pyrex pan as I do, I just stick it in the fridge after dinner to let it firm up over night.
I agree with the others about adding some fat - a nice big pat of butter should do the trick.

Fantastic Mr Fox that link was very helpful, thank you. Dahliat, I checked out the article on Cooks Illustrated and it too was great. I think I know what I need to do now and what I was doing wrong.

According to a lesson on Rouxbe.com for firm polenta you need 1 part cornmeal to 3 or 4 parts liquid. For soft polenta, it should be 1 part cornmeal to 6, 7 or 8 parts liquid. Since cornmeals vary, they cannot give an exact amount and suggest that you try different variations until you get it the way that you want it. They also say that if you are cooking polenta and it is thicker than you want, add your hot liquid, a ladleful at a time, stirring it in as needed. You can also add those tasty additions that others have suggested, butter, cream and cheese.

I think there's an old Italian saying, "you don't wait for polenta." Have your toppings/sauce ready, and plate the polenta right away. It should go straight from the pot to your dinner plate or polenta board.

I add a little butter/cream to keep mine softer. But if you intend to fry up leftovers, firm is good! Spread leftovers flat out onto a dinner plate and chill.

Good luck!

I also like to add whole milk/cream once it's done cooking to make it softer and creamier. And cheese of course!

Lots of fat has always been my answer to keep polenta setting up stiff. Butter, olive oil, cream. Although I'm sure there is a purist somewhere that will poopoo that idea. Works great and gives a nice rich taste.

Maybe drizzle some olive oil on it. Here is an interesting exposition on the basics of polenta, if you are interested:
http://www.globalgourmet.com/food/egg/egg0297/poleckry.html#axzz1AwfvHJvZ

The type or polenta or cornmeal you use depends on your preference, fine, medium, or coarse ground. One is not better than the other. They just produce different results.
You probably don't want to use grits as the process to make grits is different than that of making corn meal. Masa is probably too fine for making polenta but is great for making tortillas or the like.

another idea is to do baked or fried polenta, like the griddled polenta cakes this site is suggesting above. That way the firmness of the dish is intended.

Cook's Illustrated just covered this recently - they recommended a coarser grain (about couscous size), but degerminated rather than whole-grain so it doesn't stay too grainy when cooked. They also recommend adding a pinch of baking soda to the cooking water to help keep it creamier and reduce cooking time (about 30 minutes total, with just one stir after 5 minutes).