What would you recommend for someone who was just going to start out for sound equipment...low cost and minimalist.
For a digital piano, acoustic and electric guitar, and two vocalists?
For smaller indoor venues.
Let's leave out the electric guitar for a moment, which brings up its own set of problems. You need a PA system of some kind. This is a collection of functions: Mixing, equalization, effects (reverb, mostly) amplification and speakers. The larger the system, the more likely that all of these will be in separate units. But for simplicity, cost and portability, you'll want to get those things in as few boxes as possible.
At the level you describe, you will probably end up with either a combo PA "Head", (comprising all the aforementioned functions except the speakers) and "passive" speakers, or an integrated mixer/console with "powered" speakers.
Powered speakers are speaker cabinets with amplifiers built into them. They've become very popular. One of the pluses of having the amplifier(s) built into the speaker cabinet itself is that there is that they are frequently designed with protection circuitry that makes it harder to blow the drivers (woofer and horn driver) than it would be with a passive speaker and a separate amp. I use a JBL EON G2 as my keyboard amp. I've been using it for over a year, maybe 150 gigs so far. Various mishaps have occurred that made loud and nasty noises come out of it, but so far (knock, um,
plastic, the cases are made of plastic) ) the speaker and horn drivers have been OK.
Speakers, powered or not, come in different configurations, but for your purposes, they will consist of a woofer (low frequency speaker) and a horn in the same cabinet. I urge you not to settle for cabinets that have a woofer and "tweeters" (very small speakers that can only reproduce the highest frequencies). These leave a frequency response "hole" between the highest frequency the woofer can usably reproduce and the lowest frequency the tweeters can put out. Think "annoying mobile DJ" sound. The missing frequencies are critical for the vocal, guitar and keyboard. Be careful, some tweeters are designed to look like small plastic horns.
The most important decision you'll have to make is how big a woofer you can afford and are willing to carry. Even as a duo (I'm assuming) and without a bass player, you'll probably want to make some bass sound; the lower notes on the keyboard, for instance. 15" woofers are nice for that, and nothing larger is necessary for your use. But can you get away with 12"? That's up to you. You'll really want to find a way to play your instruments through a few kinds of speakers. A weekday in a music store, during school hours preferably, might be a good time. If you find a helpful salesperson (that's sometimes a big "if") that is willing to let you experiment in his store, I think it's the right thing to do to buy from him. You can bargain, using evidence of prices researched on the 'net, but it's poor form to take an hour or two of a guy's time and effort only to buy it elsewhere.
Larger louder groups like mine usually have "Main" speakers for the audience to hear and "monitor" speakers for the musicians to hear, at least for the vocals. You won't want the extra expense and complication yet. If you don't play too loudly you may be able to get away with putting the speakers behind you and out to the sides. This way they can serve both functions. But you'll probably want to get stands for them, to get them up high enough so that listeners in the back are not blocked by people in the front. Hey, you're good, right? Of course there'll be enough people to make that important.
Your "mixer" will depend on the type of speakers. If you get "passive" speakers - speakers without amplifiers - you'll need to get a "head" with an amplifier in it. (yes, the amp could be separate, but I sense from your original post that you are new to this, and the fewer complications and the lower the cost the better).
The mixer will need enough "channels" to accommodate the greatest number of mikes and other inputs that you can imagine using. It used to be that all inputs would be able to take microphones, but these days some are just for "line" sources (like your keyboard*). Whatever confusing wording may be used, microphone inputs have "XLR" connectors, about 3/4" round, with three little holes in them. You'll need enough of these for any vocal mikes you may need , and to mike the acoustic guitar.
The acoustic might be an acoustic/electric, in which case I'll leave it to someone more familiar with these to tell you if you can go into a "line" input, or if you'll need a "direct box", which is a separate small unit, and requires a microphone-style mixer input, but no microphone.
Get a mixer with a few extra inputs. You might expand your group, invite a friend over, want to play MP3s between sets or have an input go bad.
The mixer, or combo head, will need to have reverberation, at least. Internal effects units in mixers tend not to be so great, but you can probably add an outboard unit on later. Ask the salesman if the mixer has the required "Aux Output" to allow this. Ask someone else if he hems and haws, or can't show you where it is.
Any mixer you can use will have tone controls, Bass and Treble at least. You'll want these on ALL the channels you intend to use. This also used to be standard, but there are tiny mixers these days claiming 6 or 8 or 14 channels that have only a couple of full-function ones. The mixer may also have some sort of overall graphic EQ to shape the tone of the sound to fit the speakers and the room. This, used judiciously and with caution by novices, can be a good thing.
I won't go into microphones here. There are too many. Shure 58s are decent-sounding and don't break. Beyond that, try out some that you can afford, preferably with the speakers you want to buy. When working out your budget, don't forget that you'll need cables. But don't buy special magic "oxygen-free copper wound by virgin nuns in remote clifftop monasteries" wire. Quality construction and connectors will stand up to abuse, but any decent wire will get your signal from place to place unscathed. This especially goes for speaker cables.
Should you buy used? Maybe. There are lots of folks like you, (but not nearly as good as you, of course), who buy gear and never take it out of their basement. That could be an excellent person to buy from, the equivalent of the little old lady who only took the car to church on Sunday. But it's hard to be sure that the little old lady doesn't have a punk grand-nephew who drives the car on the beach with his slacker friends, and it's hard to know if the guy in the basement uses his system to play back Rage Against The Machine at top volume on Friday nights, for
his slacker friends. You'll want to try it out, as loud as it will comfortably go. Make sure each woofer and each horn works. Do you hear any odd noises, like maybe a speaker is about to go? Try all the channels and controls too.
When you get locally famous, as surely you will, will you just have to scrap the gear you've bought for larger stuff? Not necessarily. One of my band's singers had a small PA years ago. We're still using his 12" JBL EONs for vocal monitors. (Those things are indestructible). Another singer carries a Mackie combo PA head, just in case any part of our regular system goes down. We used it once as a bass guitar amp. The mikes will be usable even if you expand, as will the cables and stands, probably.
I commend you if you've made it this far. I do tend to get wordy. Good luck and enjoy
*the keyboard is not technically a line-level source, but that doesn't matter for this level of discussion
