BJT Class A Amplifier - Help calculating resistor values \$R_C\$ and \$R_E\$
Warning - rules of thumb being used!!

I always try and get about 0.5 to 1V on the emitter (lets say 0.75V). This means the base will be at about 1.45V and this allows you to calculate R1 and R2 (but be prepared to revise those values should the base current drawn be a little high).

You want 10mA thru the collector and this is virtually the same current through the emitter so now you can decide what Re is - 0.75V/10mA = 75 ohms.

At 10mA collector current you want collector to be half rail so that means Rc is 5V/10mA = 500 ohm. Another rule of thumb - current thru R1 and R2 should be about one-tenth of Hfe lower than Ic. If Hfe is 200 then current in R1 and R2 is about 10mA/20 = 0.5mA. 

R1 + R2 therefore should equal 10V/0.5mA = 20 kohm.

R2 needs to be smaller than R1 to develop about 1.45V on the base and on a 10V supply I can immediately see that R2 being about 2k9 would take 0.5mA to develop 1.45V. This leaves R1 as 17k1 ohms.

I'd use 3k0 for R2 and an 18k for R1 for practical reasons of obtainability.

C3 depends on your lowest frequency input and how much voltage gain you need. Without C3 there is a voltage gain of approximately Rc / Re = 6.667. If you want more gain use C3 but be prepared to put a small resistor in series with it to linearize the gain over the full desired bandwidth.