Is there a workout I can do to make me tap faster?

Well, speed is not the only thing you should be aiming for in tap. Even though you will need some speed to do some choreographies. Musicality, swing, feelings and your own style is much more important than speed. (I think :)
But if you do feel that you want to get more speed to be able to execute some steps to faster music, then I recommend working with Metronome. You can change speed to your liking (don't have to look for songs that is right speed for you) Start slow, then slowly pick up the tempo. Then once you get to the speed you are reaching for. Then it will be good to dance to a music that is in the tempo and style that fits the steps to get the feeling how it fits with music.

Should a beginner start out with a cheap tap shoes or invest in a top of the line shoes?

I would say start with inexpensive one first. But make sure it's right size for you and feels comfortable. As a beginner, you don't have technic enough to prevent yourself from injuries. So it is important to have shoes that feel comfortable to you.
When I started tap dance back in Japan, my teacher did not allow students to buy tap shoes for about 3 months. We had to get okay from him to buy one!!! So we were in sneakers for the first few months. Well, that is little extreme. But it certainly made me work hard so I can reach my first goal of putting tap shoes on.
Once you get basics and feel like you want to continue, then I would recommend you to get better shoes. It does make difference in sound.

By learning to play music would that make me a Better dancer ?

Yes, yes and yes! You should want to know what you are dancing to.
If you ever have a chance to learn music, it will help you a lot in your dancing.
If not, just listen to a lot of music that you want to dance to and feel the music in your mind and body.

One thing a performer always need to keep in mind is ?

For me where you look is very important.
Your Eyes tells story more than anything you do on stage.
Wether you look at audience or look at something else or even close your eyes.
It will be your choice but it changes the feel and dynamic of your performance.
Eye contact! with your audience? with your musicians? with other performers on stage? or with yourself?
It is how ever you like it to be, but be aware :)