I don't know what to work on between classes

(The above statement assumes either the lack of assigned homework by the teacher, or you wanting to do more after completing such homework).

Some of you will just get by in this class - that is, if you can't invest enough time between classes for practice. 
Some of you will love the challenge and will want to do more; in this case, your job is to take your music training into your own hands. 

It (the job) consists of a number of areas to which you'll need to dedicate separate time chunks:

1) Seeking out musical materials that are composed/arranged for piano, and figuring out practice strategies as you try to learn how to play these materials. 
This area is huge and involves the 
-ability to read music
-using your body/fingers well
-staying focused on learning short chunks (think goal-oriented approach).

-Using the assigned text book
-Using supplementary materials listed on the syllabus
-Finding your own books online (amazon, sheetmusicplus.com, etc) - this will depend on your preferences (style, composer, arranger, etc.)
-Sites like IMSLP.org

2) Listening to plenty of different music - you cannot be good at something unless your brain is sufficiently steeped in that field to where your ability to think on your own (musical) feet is reinforced by your 'inspiration baggage". 
You'll never have enough self-generated material to push you beyond your current capabilities; you need to hear and observe people who do it at a more advanced level than you.

3) But to learn it all, you need to immediately start creating - improvising, then writing down what you come up with. This will train your ears, will force your fingers to work so as to execute your creations, and allow you to become comfortable with musical notation.

4) Sooner or later, you'll also need to figure out ways to practice performing. That is, making a musical statement on a piano for someone else. In the beginning, it could be you yourself pretending to be an audience member, while your hands are busy being performers. Later you should involve good supportive people, like friends, family or pets.

A lot of you will want 1) to be your primary goal in this class - simply learning how to pick up some piano music and learn to play it quickly. I can give you advice there - and the book contains plenty of material to give you something to sink your teeth into.

If you are truly going for the ultimate use of your time in this class, you should start a "practice diary" (I myself use googledocs or excel for that), where you input what you are working on, what approaches you are trying, and the amount of time you are spending on practice. 
It's best if you have this diary handy (think smartphone with a good app, or the old-fashioned booklet and pen) where you can write down any piano/music-related thoughts as they come to you.

I don't know how to choose a piece of music to work on

Inspiration comes from sitting down (or standing up) and doing the step-by-step routine.
In the case of choosing a piece of music, your first task is to surround yourself with sheet music that you can choose from. Additionally, you can listen to a lot of piano music using Youtube or whatever you like.
Once you have chosen a group of pieces that are at your current level of ability (you'll figure out soon enough whether that's the case or not), your next step is to start sight-reading. You look at the notes, and you start trying to play them. If this is tricky - which it will be if you have never done it before - do one hand at a time.

Once you have gone through a number of pieces that way, you should be able to gravitate towards one of them as your "preferred" piece. That's when you pull out all stops, and start working on it using all the different strategies you know about from life experience, and those that we go through in class.

If you stick with that routine, of having a library to choose from around you (whether as paper books or on your iPad or laptop), and make sure to sight-read at least one part of one piece during your practice session, you will accumulate enough knowledge of the literature to make the piano piece choice easier for yourself.

If you can't find any music - use anything in your classes's textbook. Those pieces are designed to be fun, straightforward.

I'd like to learn to play fast pieces without stumbling

If you are at this point, it should imply that you are able to play these fast pieces slowly without stumbling.
If that is the case, there are a number of methods  you can use during practice that will help you to "speed up" to your desired tempo.

1) Choose a concrete passage to work on - just a couple of measures with a clear beginning of something and an end. If the flow of fast notes is constant, there are still harmonic/cadential points that give a sense of "arrival". So, choose.

2) By experimenting, find the speed that is as fast as you can go without a) feeling tense, b) stumbling in any way. Remember that speed and stick to it. If you stumble, reduce speed further.

3) Learn to speak (with your voice) the names of notes of the "fast moving" part - you might need to slow down even more. By doing this, you will disentangle any "finger confusion" in your brain.

4) Use "Goal Oriented Backward Stepping" method to focus on very short segments at the very end of this passage. Learn to play them at the fast tempo immediately. Work backwards, and shift "the goal" as soon as you start to mess up.

Combination of these approaches, should give you a much better handle on tricky fast passages, or at least make it clear what causes you to fail.

NB: Do not over-practice till you are tired. Your job is to achieve a sense of accomplishment in the smallest possible fragment. Then take a short break, or do something completely different.
NBB: Do not expect obvious results in one practice session. You need to do good, solid work, then sleep on it.
NBBB: Repetition is key. Every time you repeat, discover new things. Learn to feel carefree (as you play something that seems to demand "lots of care"). Focus on being "lazy" - doing the least amount possible to get the results you want.

Some other concern? Send me an email