What kind of hay you get depends on what is available in your area. Here we have timothy as far as grass hay goes. At least that is the most common.
What kind of hay you get depends on if your horse is an easy or hard keeper. Some horses do great on grass hay alone, some horses need something a little richer to keep weight on, or a supplement like grain.
I feed alfalfa right now because the price of grass hay here is ridiculous. $7.50 - 8.00 for a 50-70 pd. Bale! I'm able to get a 130 pd bale of alfalfa for $14 from my trainer's dad. It lasts longer because I don't have to feed as much due to the high energy content, and oddly enough its pretty much the same price as the hay (if not a bit cheaper) when you measure the amount versus the price. Alfalfa is supposed to be more expensive than grass hay! Stupid economy....
In any case, some horses may get too hot on alfalfa. So my advice would be to experiment and see what your horse does best on. Buy a couple of bales of whatever type of grass hay is prevelant in your area, and a couple of alfalfa and try her on both. If budget is an issue, generally grass hay is easier on your pocket (at least it should be.)
Whichever you get, make sure it smells good, and is free of mold, dust, and trash. Just like everyone says. Just because grass hay isn't green doesn't mean it is no good. Just make sure it isn't as yellow as straw (which is not suitable for eating, and is used for bedding material.)