a better source and a competent soundguy can make bad stuff sound good, but this is simlpy false. the transducers (mics and speakers) are vital to the overall quality, and make a huge difference, maybe the biggest difference.
cross over the subs at 100Hz, adjust their volume to just add in what's missing in the mains, turn the lows down on everything except the kick and the bass, turn all the instruments the hell down and turn the vocals the hell up, and you should be OK.
Yeah, better speakers can sound 100 times better, but they are usually out of the range of a small gig band. A good sound guy can even coax a decent sound using a box of old 58s and Peavey SP-1's for a small live show.
Definitely start trying to cross-over at 100. If you don't have a crossover, the board may have a "sub out." If not, another trick would be to take an extra out form the board (or run mono and use left or right) and EQ it to mimic a crossover.
Your outboard EQ is almost as important to a clean sound as your board so be careful with what you use. I've always advocated removing unwanted frequencies and avoiding boosting anything because that potentially adds noise.
The suggestions about slotting the instruments and starting with the stage-only mix are dead on.
And I agree totally that you shouldn't just go out a get a compressor. They take a bit of study and practice to use them properly.
Learn all you can about gain structure starting from the mic preamps on each channel of the board and keep your signals clean and optimized for headroom. Mackie and Allen & Heath have literature on line and in manuals that is easy to follow.
Read about mic technique, patterns, and proximity effect and try to show your bandmates how "powerful" a mic sounds when you use it correctly. This will allow you to have the mic gain lower and not pick up everything else in the vocal mics. It will also give you more headroom before feedback in the monitors.
Aim the monitors at the back of the vocalists ears (from the front, of course, not straight up into the mic.
Keeping the band at a reasonable level on stage is uber important.
Keep your eyes open for an old cheap Peavey Gatekeeper or some other multi-channel noise gate. You can use it to start tightening up the drums and eliminate bleed. I can't speaker on the Behringers because I never tried their gates. You should be able to find something to start with at around $65-$80 in a single rack space with 4 or 5 gates. Entry level dbx gear sucks, so save for the good stuff or hire a sound company when you get to that point

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Just my 4 cents... Have fun!
Mark