When it's ready to bloom again, you'll see a blunt nub that looks a lot like the root start to form near the base of the plant, but instead of growing down, it'll start to grow upward -- that's your flower stalk.
Depending on your light situation, the stalk may or may not need to be supported -- did your blooming plant have something supporting the flower stalk? I try to give the pot 1/4 turn every day or so, and also turn the flower stalk AWAY from the window to encourage it to grow vertically. This makes it easier support upright later on. If the light it gets is on the weak side, the flower stalk will be somewhat floppy and will not be able to hold up when the flowers bloom to weigh it down.
* Note * I don't actually put a support on the flower stalk if I'm tying it to a bamboo stake or a chopstick until the flower buds start to form. I don't know if this is correct, but I'm always afraid of accidentally damaging the flower stalk or where the buds will grow out. I *have* used the wire with a loop on the end that came with my orchids which only touches the stalk at the wire loop to train a growing stalk to grow in a desired direction, especially if it's a wayward stalk that's trying to grow sideways.
I live in a climate way different from yours so it's no indication, but my 2 flowering age Phalaenopsis are starting to send up flower stalks now.
Another reason for not cutting the flower stalk right away -- just like shadowsmom said -- is that the plant will sometimes develop a baby plantlet on the flower stalk that will eventually form its own roots and can be potted up separately. I have a new little plant with 4 2" leaves (about 4" in diameter) now.
(I *would* have 2 new little plants except that I accidentally let one dry out during the summer -- the little ones need a bit more TLC
I occasionally water my orchids with what I call "used coffee ground dregs water" -- I use a personal French press to make coffee so I add filtered water to the carafe with the UCG and water with the *very* weak coffee, and let the UCG pour into the orchid pot.