Gecko always seems to give good, (ahem) sound advice. Hard to find fault with his recommendation this thread, either.
I have a lot of info below- whatever is in
bold is of most importance, if you need a quick read, go to that.
Gecko's mixer suggestion has a good number of inputs- I would go with 16- no fewer.
You don't mention if you will be using a snake (typically 16 cables that bring the signal from the stage to the sound booth or tent, 4 cables that bring it back to speakers- thus a "16X4" snake)
but you might want one if your budget allows. A snake allows you to HAVE a sound booth or tent, otherwise the sound man is on stage or just off in the wings- not the best place to hear what the sound is like for the house.
Most snakes should NOT be used with a powered MIXER, as it can burn out the 4 "returns. The only solution is a "hot" snake, and they cost much more.
IMO, the powered mixer Jim Lad is less than ideal for three reasons:
1. Incompatable with snakes you might afford.
2. Only 8 inputs with pre-aps (XLR inputs, for mics)
3. Behringer has a bad (and well-deserved, IMO) reputation, esp. with mixers.
The three options you have for the mixer/amp/speaker are:
1. Powered mixer (amps in it)-passive speakers.
2. Passive mixer-powered speaker (speakers have amps built into them.)
3. Passive mixer-power amp (located as close to the speakers as is practical)-passive speakers.
IMO, the last setup is the most flexible but the most complicated. As you are a NOOB,
setup 2 is probably the best for you.
Forget crossovers, regardless of what setup you go with- you will be out of your league. Get some experience under your belt, first.
Same with subs- added complexity/headaches you don't need right now. And don't EVEN get a compressor right now- you will be flabbergasted by "all the knobs and buttons" on it.
An EQ is primary used to control feedback- there is always a small one (3 to 4 bands) on all mixers, which should do you for now. If you must get one, you should go with a 31-band X 2 channel unit, which will probably bust your budget. Each of these devices- EQ and compressor- will necessatate yet another cable, which means a little more money spent.
KISS (keep it simple, stupid!) and KIIB (TRY to keep it in budget.)
Another thing you don't ask about is stage monitors. Most all mixers have one, two or more monitor outs-
two is enough for what you will be doing. Use two of the snake's returns for the monitors-
two powered, 50-100 watt monitored speakers will do you fine.
One piece of advice Gecko leaves out is
about buying mics. Shure SM58's are the industry standard-
not cheap, but, well, not cheap. Their reliability, durability and sound make them
worth the price. Shure also makes a SM57, which is a bit more an "instrument" mic (in front of amps, horns, etc.)- whereas the 58 is more "vocal-" but 58's tend to work better as inst. mics than 57's do as vocals. Seindheiser makes good mics, too, and they tend to be a bit less expensive than Shure, but IMHO the Shures are worth the extra coin. Be VERY careful you don't buy Shure counterfeits- do NOT buy mics via eBay, or used- you simply lack the experience to be sure you don't get burned. If you decide you need semi-disposable mics, NADY makes some that, although perhaps not as good or robust as Shures, are much cheaper.
If the users of the mics are inclined to not respect them as the precision instruments they are, you might go with those cheap NADY's. One stand per mic is fine, but get extra mic cables- they take a lot of abuse; have a spare or two on-hand, you will thank me later.
Get a pair of SPEAKER STANDS for the house speakers.
There is a WORLD of difference between SPEAKER and INSTRUMENT cables, although they can look a lot alike. They are NOT interchangable.
Here is the basic signal chain, which includes everything you originally said you thought you needed. For things you don't get (compressor, sub, maybe EQ) just take them out of the list: ( ">" means "connected to.")
POWERED MIXER SETUP (not recommended, IMO): Mixer >expensive hot snake> speaker cables > passive speakers. Compressor and EQ connected to one of the mixer's effects loops (see what I mean by complexity? Already we are talking about things you have never heard of...)
PASSIVE MIXER/ POWERED SPEAKERS (recommended for your present skill level): Mixer >EQ > compressor (EQ and compressor can be on effects loop, or in-line) > snake > instrument cables > speakers.
SEPARATE POWER AMP(S): Mixer > EQ > compressor (EQ and compressor can be on effects loop, or in-line) > snake > instrument cables > amp(s) > speaker cables > speakers.
Run house speakers in MONO, not stereo. Lots of good reasons, which all boil down to easier life for you and better sound for everyone in the audience.
Similarly, one monitor feed is best, unless you have REAL good reasons for more (prima donna on stage, real problems with some musicians needing more/less of other musicians in "their" monitor)- and then, NO MORE THAN TWO. Monitor mixes are the most mind-boggling, you do NOT want that kind of mess in your life, right now.
One more thing about EQ's: They tend to be used to "fix" the room acoustics, but you are opening a real can o worms, there. Use them, sparingly, to fix feedback, and let the room be what it is.
One more thing about compressors: Best used VERY sparingly, too. I use them to protect my speakers from a belter vocalist who keeps asking for more of him/her self in the monitors- and learned that lesson by losing a monitor.
When buying, you will be better off with a PRO AUDIO company, not a general music store like most mom-n-pops or Guitar Center. You may pay a little more, but you will get better advice, less probability of being sold something that is pretty much useless to you, much better service and support after the sale, and probably a more generous trade-in policy if you decide to up grade. You might even get, if you ask for it, someone to help you set the system up the first time you use it.
Do me a favor- print this out and bring it to the pro audio seller you go to- ask the sales person to read it, and give you his opinion. Let us know what he did, and did not, agree with.
Good luck.