Designing a multi-stage BJT amplifier
Your design is a very basic and might work but since it has no feedback it is not so predictable in how it will behave in practice.

The "proper" engineer's way of designing this is by use of feedback. You basically make a crude amplifier with a high gain and enough bandwidth and use feedback to get the gain you actually want.

Unfortunately this design procedure is not something which can be explained in a few sentences. I learned this in a course which took several days and included a design assignment.

I found a University course here that should explain this method, that's 70 slides to get you started ;-) 

If this is too much of a stretch in the time you have then just remember this and come back whenever a more challenging amplifier design task pops up.

Another option altogether would be to use an opamp in a feedback configuration. If you want to know more about opamp circuit design look here: Opamps for everyone