Is what your saying you need 4 speakers for $750??
You won't get 4 good speakers for that unless they are very small passive. Then you need a power amp which will cost that amount too. For that price point you are not looking at the top brands at all. Example that's the price for just one Yamaha , QSC or JBL powered speaker. Have you not looked on the Sweetwater or MF site yet??
For sure you will need powered speakers. The best bang for the buck is Alto. Excellent sound and so far hard to find a negative review. The 10" are fine too if just for simple acoustic music. I've had mine for more than a year now and I'm very pleased. Lots of clean power and even though they are only 12" they kick out a good low end. I'm not a fan of 15" speakers anyhow.
https://www.altoproaudio.com/products/ts212 That's not a good idea to put speakers at the rear of the room unless you understand how to set them up with a delay, very advanced skills needed for that. If you can fly the speakers or just get them up there on stands.
Shopping used or demos in music stores is a great plan-I picked up a pair of Mackie 10" powered speakers that had been demos for only $150 each. New they were $450 each
Analog mixers are very affordable these days. My top pick last year was a Yamaha MPG 16. That handled a 5 piece band no problem. I chose it over the others because of the feature set and our needs. Every situation is different. Like how many monitors and Aux you need.
Make sure to count the XLR inputs and faders. They tend to call mixers 12 channels when It's really 8.
This is the mixer I would buy this time because I'm in a 3 piece band now.
https://www.sweetwater.com/store/detail/Sig12MT But I will probably go with a Behringer X12 air. Which is worth looking into if your tech savvy and have an iPad.