24GHz Radar : IF amplification and strange mixer output
With amplifiers right at the output of the mixer, your noise is dependent on your bandwidth (\$kTBF\$ where B is filter bandwidth, F is noise figure etc...), so in order to not blast your receiver with a bunch of noise or saturate the amplifier, you would want to use a BPF to only select the bandwidth within your range, which also serves the purpose of eliminating spurious signals from the mixer. 

I'm not sure on the second amplifier, it may be there just to provide more gain (cheaper components) and you need another BPF to filter again before that amplifier so you again don't amplify a bunch of noise. Your LPF there at the end is probably used to eliminate the IF mixing component. Your output has a Doppler frequency component and an IF frequency component, where your Doppler component is order(s) of magnitudes smaller than the IF frequency.  

You are correct in saying the your mixer output would be your signal frequency minus your LO frequency, but that just gives you your IF frequency. Your information is stored in a different component, the Doppler frequency. It would be helpful to know more about your system (such as LO frequencies and the specs of the components).