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Step 1: Get Into Your Dance Frame
The leader holds his left arm up, elbow almost at a right angle. His hand should be approximately at his partner's eye level. His right arm holds his partner, with his hand at the base of her shoulder blade and his arm up to support her arm.
The follower takes the opposite position, with her right arm up so her hand is in his, and her left arm above the leader's right, her left hand just above his bicep.
Align your bodies so that your right sides line up, not the centers. Keep your feet a little less than shoulder-width apart.
Do not intertwine fingers; this is important if you want to learn how to spin the follower around. Fingers break that way.
Step 2: Push Your Partner Around
After you've mastered having the leader go forward and the follower backward, switch that around. This time, the leader should try to move backward, and the follower will move forward. This is a key point for the follower, though: he is leading. Don't lead him. This means that the follower should always start and finish a step just slightly after the leader. This also means that, contrary to popular belief, if the follower is stepped on, it's her fault - she didn't move as far as the leader tried to make her. Unless, of course, the leader is just being mean and taking steps that are too big for her to do.
Small steps for now, folks. Somewhere around 18" is comfortable for most people.
Now that you can go forward and backward, try going sideways. The leader will move his left foot to the left, and bring his right foot up to his left (remember, keep your feet a little apart); this should force the follower to move her right foot to her right, and bring her left foot into line. Try going the other direction too.
If you really want to practice this part well, blindfold the follower. Leader, mix up what direction you go.
Step 3: The Basic Waltz Box
There are two halves to a waltz box. Get into your dance frame. Leader, you will start the box by stepping forward with your left foot. Now, move your right foot to the side. Bring your left foot next to your right foot. When you have music, each of those three steps will be one beat of a measure. That's the first half. Check the text boxes on the images; they should help get the order right.
The second half starts with the leader going backwards with your right foot. Then step sideways with your left; and bring your right foot back into position. Voila! You've completed a box.
The follower will end up doing the exact same sequence of steps, except she starts with the leader's second half of the box - moving backwards with her right foot first - and then does the leader's first half of the box as her second half. So follower, use the right image first, and then the left image.
This box is key. You can do a full waltz dance just by repeating this box (although you might get bored.) I'll teach you how to move forward next.
This video has some good music to practice with at this point. You should take a step on each beat, and you should alternate the foot with you step with on each beat. If you're having trouble finding the beat, watch her play the guitar: she plays one note with her thumb, then finishes the chord, and then she strums it twice. That first pluck is the first beat, and the two strums are beats 2 & 3 of the measure. You'll find it a little fast to start with, but after some practice you should be able to dance a series of boxes while staying with the beat.
By the way, for every ballroom dance I know (and I know a bunch) the leader always begins the dance with his left foot, and the follower always starts with her right.
Step 4: Moving Forward
In order to move forward, start off with the first half of the box. Now, instead of going backward with your right foot, go forwards (backwards w/left for followers.) Then do that funny diagonal step forwards with your left foot, swinging it in like a forward step and then left like a sideways step before putting it down; then complete the move by bringing your right foot in.
You can repeat this for as long as you want to move forward. Here's the link to the video again to practice with some music.
Step 5: Turning Corners
You're going to start with the same step you've started with each time. The leader will step forward with his left foot, and the follower backwards on her right. Now, though, it changes. Don't take step 2. Instead, pause for a beat. Then, for step three, put your weight back onto your right foot -take a step backwards, but since you never moved your right foot, you're simply stepping back onto it, and bringing your left foot back into position. This lets the follower know that you're about to turn.
On the first beat of the next measure, take a sideways step with your left foot - and turn your body with it so you're facing what was your left. You'll need to practice this a few times. You will find it's easier if you rotate your right foot towards your left at the end of the forward-and-back step I described above.
In the image below, the text boxes over the arrows are the actual steps; the one on the foot is the beat where you pause without stepping.
For your next step, beat 2 of the measure, put your right foot to the side of your left. Then move your left foot to the right, so it's next to your foot. Now you can go back to doing forward steps. There are (physically) other ways to transition from a turn to a forward or box move, but remember: forward and backward steps should be on beat 1 of any measure, not beat 2 or 3. This particular transition will keep you on the right beat.
Try this move without music several times. Combine it with some forward moves, both before and after, also without music a few times; then, try it with music.
Step 6: You've Got It; Keep Practicing.
If there are people in your way so you can't move forward, use a box step to wait until they get out of the way.
Dancing will be easier if you keep your frame in place. Your arms are your connection to your partner; make sure it's a strong connection, and it'll be easier to lead and follow properly. (In other words, you won't step on each other as often!)
When you take a forward step, put your heel down first, and then your toe. Putting your whole foot down at once can add jerkiness. Similarly, when going backwards, put your toe down and then your heel.
For all steps, put your foot where it should go, and then move your weight onto it, rather than doing both at once. This will help smooth everything out. Waltz is supposed to be graceful and smooth; you're dancing, not walking.
With that said, it can really help you use the correct foot at the correct time by thinking of it like walking: always one foot, then the other; never the same foot twice (except at that pause in the turn, but that's a special case.)
You can also make it look smoother by curving your forward steps. You don't have to go straight forward all the time; in those Disney movies, you can see it looks like they're spinning around. It just means you won't be closing the box all the time. When you step forward on your right foot, rotate it more to the right and bring your body into line with it, like what you did with the turn, except turn 30 degrees instead of 90. And remember you're still going forward.
Google is your friend. Lots of ballroom competitions are filmed, and you can find those videos to see what it's supposed to look like. This was a waltz; find videos of people at the Beginner pt Bronze level, and you'll see the moves I just taught you. (Silver, Gold, and Open are more advanced.)
I intend to add a follow-up to this, with a couple more moves like an underarm turn so your dance is more interesting. I also hope to post instructions for other dances, like the cha-cha.