You can't beat the QSC K-12's for power and quality, but they aren't cheap. They are lightweight and compact for the power they pack. You would need tripod stands with these.
A small mic mixer can be bought these days for just over a hundred bucks.
There are knockoff copies of the K-12's and also the Mackie 450's. Behringer sells some for quite a bit less. These compact, powered speakers are the way to go for small venues.
If you want to go on the inexpensive side, there are a number of copycat versions of powered speakers with a 10-inch or a 12-inch speaker, a small horn, and built-in amplification. Some of these even include a simple input mixer on the back side of the speaker. It is customary to use two of these, mounted up on a tripod stand, but nothing says you couldn't get by with one.
Behringer makes one for about two hundred dollars apiece, but you need a small mic/line mixer for your mic and guitar. That would be about the least expensive for new equipment at an entry level.
The QSC's are what I use in my club band. We turn them way down at several venues, but they could be played at higher volumes. They are loud if you want them to be, and extremely high-fidelity sound. If you played in a really loud band you might need something bigger and more industrial-strength, but that would run you a lot more and require a truck to haul it. Big systems take up a lot of space for storage and require bigger stages.
I bought the QSC K-12's because I often set up the sound system by myself and I can lift the K-12's to put up on the tripods without help. They have plenty of power and volume for the venues I play at. That's important. I have Gator tote bags for them that are on wheels with luggage-style collapsible handles. That makes getting them to and from the vehicle easier.
Get something that sounds good to your ears, is within your budget, and will last for a good while. Make sure you can handle it, get it up on the tripods, store it, and move it around. If you're thinking of getting into a band, the QSC's would work, so would Mackie 450's, or copycat versions like Behringer.
You get what you pay for, but don't get taken, either. Shop around before you commit to buy. Get a good mic like a Shure SM 57 or 58, or a Sennheizer E 835. An inexpensive mixer these days will often include built-in reverb and/or echo, which can be a plus.
Compact, powerful systems are good, because you can transport them in a minivan, or possibly a hatchback or station wagon. Big equipment requires bigger vehicles and bigger roadies.
The SM58 is the workhorse mic for musicians all over the world, as is the SM57.
I use them all the time, but for my main vocal mic I replaced the SM58 I was using with a Sennheizer e835. It has a little more extended high frequency response, which I personally prefer. If I could, I would get one of those Neumann live mics, like the KMS105 with 20 - 20,000 kHz frequency response. Since money doesn't grow on trees, I depend on my Sennheizer.
High Quality sound gear should have the best quality input you can afford. Garbage in, garbage out.
Diamonds in, diamonds out.
I think you need to put an XLR cord from your mic straight in to the XLR input on your Tonematch mixer.
The quarter-inch adapter may be making the Tonematch think you are applying a line-level signal instead of a mic-level signal, which is much lower in level and requires more preamplification. Also, the adapter may not be properly matching the impedance of the input channel.
Work with the trim knobs on the input channel to find the strongest signal just below where you get an occasional red peak indication on the led with your strongest (loudest) vocal you will be singing. It's usually OK to see an occasional flash of red, but this looks like a digital mixer, so red might mean ugly. Trust your ears. Get it sounding good.
Then use the volume knob on the channel, as well as finally the master volume to set the level.
Your Tonematch should provide more than enough gain to drive your Bose system with an XLR mic-level input.
Echo and reverb are standard, and compression is nice, once you learn how to use it. You also want to have E.Q. to help get the sound you want.
Oh, yes, and did I mention enhancers and aural exciters? Those are used to make phase-corrections in reproduced sound, and the result is a more high-fidelity crispness in the sound. You have to hear these to understand whether you want to use them or not.
You can get all-in-one "channel strips" or "multi-effects" boxes that include several of these tools.
Don't go too gear-crazy. There's no end to it.
You don't want to overdo the effects, but they can be used to put a "sheen" on your finished product that makes it sparkle a little more.
Usually you want to use the effect by adding just enough to where you can barely hear it, and then back it down a notch or two. Subtle effects can be very effect-ive.
I generally use a graphic equalizer, about a 15-band, to find feedback frequencies and reduce them.
I used to use feedback destroyers, but they tend to filter out too much of the sound and make things sound suppressed.
I'm not familiar enough with your system to know where you would use a graphic EQ with your Bose system.
One way to reduce feedback is to put the speakers forward of your mic. Unfortunately, that means you won't be able to hear it as well as the audience will. The demo video of your system showed the users talking about how they just set the speakers up behind them and used them for both monitoring and for the audience to hear.
Everybody knows (or should know) that if you put a mic in front of the speaker that is amplifying it, you will get feedback, the louder the system is turned up. Feedback is nothing more than the mic picking up its own signal from the speakers and amplifying that speaker sound through the mic, over and over again until it howls and screeches. A graphic EQ can help you to find the frequency that is most sensitive to the feedback, and to attenuate (reduce) that frequency so that you can make the other parts of the sound more audible without the howling part.
Mic technique 101: Sing directly into the mic, very near to it, but not too close. About a quarter-inch away, directly in line with the axis of the mic is good. When you sing a really loud, powerful note, duck away slightly, to have mercy on the audience's ears. Loud, High notes hurt people's ears more than lower, more moderate notes. Don't hurt people's ears.
The closer you are to the mic, the less you have to turn it up, and the less likely you will get feedback at a given volume level.
Don't point the mic at the speakers.
They are saying much of the same information I said above, such as close mic techniques, and finding the frequency that is causing the most feedback and attenuating it.
I have heard amazing sounds come from small Roland Cube amps. I wouldn't mind having one myself.
You want to travel light when busking, so again the street cube would be a good choice.
If you use it as a monitor behind the Bose system, remember to have it forward, pointing back at you 9and the backside of your mic), not behind you, pointing into the frontside of your mic. That will help to prevent feedback.
The Sennheizer has extended bass response AND extended high end over the Shure. They are both good mics, but my Shure stays in its case as a spare, and my Sennheizer is always on my mic boom.
I have a Crown headset mic that is a condenser. It sounds quite good. I do a lot of my practice work with it.
For live work, I prefer a mic on a stand, so I can vary the distance on stronger notes. With a headset mic, the mic is always at the same distance. I have a "coughdrop" foot switch for the headset mic. You need that in case you have to cough or talk to someone while wearing a headset mic, and don't want to amplify the conversation or the cough.
Dynamic mics like the Sennheizer e835 do not have quite the extended lows and extended highs (20 - 20,000 hz) that most condensers have.
Most condensers are not as rugged as dynamic mics, so the dynamics are more frequently used on stage, where the handling of the mic might be rough.
The Sennheizer e835 has slightly extended low and high end frequency response for a dynamic mic, which is exactly why I bought one. A friend bought an extra one and brought them to a gig so we could try them out. I bought mine from him that night. I was sold on it immediately, and haven't used my Shure SM58 for my vocals since then.
In a studio setting, I always use a condenser, unless I want an up-close mic sound. Then it's back to the e835.
Everyone I know who has purchased the Rode NT1A likes it very much.
The 58's are good mics. I have several of them. The Beta series is even better than the original 58's. Shure mics have been industry standards since I was a kid.
Sennheizers used to be clean outa sight, cost-wise, but now they have some good options for under two hundred dollars. Neumann has some fantastic mics, but they sure are spendy.
The Rodes are a nice option when you want to move up but can't afford Neumann.
AKG makes good microphones as well.
So you can find a lot of choices that didn't exist back when we were using those old square mics like you see in the shots of Buddy Holly and Elvis. They even make retro versions of those with new-style high-fidelity capsules inside.