Cherry MX Red switches are a lot like Black except they require less force to depress. A key point here is that they do not have tactile feedback (no bump when you press down on the keys).
Cherry MX Blue on the other hand are tactile, so you will feel a slight bump as you depress the key to let you know when you've reached the actuation point. Lots of people like having that tactile feedback for typing. However it can be a hindrance in gaming (you can't double-tap a key as quickly, for instance). These are also known for being quite loud and the keys require a moderate amount of force to depress.
I'm actually using a Cherry MX Brown keyboard, which is sort of halfway between Blue and Black - the tactile feedback is subtler but it makes less noise (but is still quite noisy compared to a membrane keyboard) and doesn't require as much force to press the keys. I'm really enjoying it and the transition from membrane to brown for me was pretty seamless. It's great for typing and gaming.
As for your last question, just having a mechanical keyboard will not necessarily increase your typing speed. What can increase your typing speed is if you get one with tactile feedback (Cherry MX Blue or Brown) and you essentially learn how to use that feedback. Essentially you train yourself to stop depressing the key when you feel that slight bump and move on to the next key - now you no longer have to push the key down all the way and that is what can help improve your typing speed. Another effect of this, that is probably more important to most people, is that it reduces finger strain since you are doing less work with your fingers.