If the powered mixer has enough power and outputs to power your speakers, then you won't need a separate power amp, since the mixer has built in power amps. Or, if you wanted to be able to have more options at hand, you could get a non powered mixer and a separate power amp. That way, you can get the mixer based on your mic and input needs, and get a power amp based on the speakers you get.

This will probably be over your budget, but this is just an example of how that would work....

>Yamaha MG16/6FX 16-Input 6 Bus Mixer with DSP<--->dbx 223XL Stereo 2-Way/Mono 3-Way Crossover<--->dbx 215 Dual 15-Band Graphic EQ<--->Peavey PV 2600 Power Amp<(for subs)--->Peavey PV 1500 Power Amp<(for mains)--->2-Peavey PV 118 Single 18" Subwoofer<--->2-Yamaha C112V 12" 2-Way Club Concert Series Speaker<. That should give you a PA that you shouldn't outgrow too fast. (I didn't include mic's, and cables, and such)