Hello, CiCi

Because it is so long ago, I can not really answer your first question, but I can give you an idea how long it has taken some of my pupils to learn how to "post" to the trot (or "rise" to the trot) and to canter.
Well, it depends on a few things:
1. On how comfortable and relaxed you are on the horse in motion
2. How co-ordinated and balanced you are on the horse in motion
3. How much feel you have for the motion itself
4. A reliable horse
I normally start beginners on the longe so they don't have to worry about the horse (because I am controlling him).
After explaining the basic position (up-right seat), we start doing numerous balance exercises in walk, e.g. standing up in the stirrups, rolling forward onto knees and thighs, shifting upper body balance without moving legs etc.
Once this is going well, I talk about the trot motion and explain what is about to happen under the rider. This is usually preparing them well as to what to expect: A "bumpy" two beat movement which has a RHYTHM.
1/2/1/2/1/2/etc.
I ask the rider to hold on to the pommel (or balance strap) to start, so (s)he can pull her/himself into the saddle for the first trot strides to feel the 2 time rhythm of the trot. Once the rider starts to relax, I as him/her to stand up in the stirrups (like we did before in the walk) until (s)he has found the most effective lower-leg position to keep his/her balance. Once we have a relatively still leg I ask the rider to gently go down into the saddle and "bounce" out of it straight away, with No. 1 meaning "down" and No. 2 "up" with the rhythm of the stride.
In other words, I can teach a rider to do the rising trot (posting)in one session.
Learning to control the horse (direction and speed) will be dealt with in this and all other sessions - in walk to start- later in trot....and canter. Provision for this is how well the rider can maintain a steady contact and body balance.
(Independent hands come from good "balance". I have found longing lessons for finding balance on a moving horse most valuable - NOT ONLY FOR BEGINNERS!).
Much the same for canter.
To "cope" with the motion of trot and canter, your body and mind has to understand and experience it first in controlled circumstances (e.g. on the longe).
Once you can comfortably balance your body WITH the movement then you can start influencing it (the movement).
In other words, you can "learn" how it feels to post to the trot or to sit to the canter, but it takes a little longer to "ride" it effectively, being in control of it. This will come with experience and good instruction. And it's true when they say: You can learn to ride a horse only by riding a horse!
To your 2)question: If there is anything you can do to prepare yourself for dressage...
Of course. You are already doing it! There are many good and informative dressage sites on the net, there are countless good books and videos. Explore as much as you can. Also go and watch lessons at your local riding school and be a spectator (or helper) at competitions.
So, you are on the right track!
As to 3) and you first dressage test.
Well, once you are able to ride all the required movements, make sure you LEARN the test, so you know exactly where you have to go and in which gait.
Practise with your instructor the components of the test (careful not to do the actual test too often, as the horse will learn it as well and will anticipate.....)
When you compete for the first time, make sure you see it as an experience, don't expect too much (if you are well prepared, you will be surprised how well you can do!). Have fun and never forget why you are competing: For the love of riding and horses - not for the price!
Hope this helps,
Cheers,
C.