Big subject. This and any other answer can only be an introduction.

You MUST define "cheap" as you see it.
What country?

Your lens will have an extremely major effect on the result. See below. A P&S can meet your spec but you really want a DSLR.

In high light situations some quite basic P&S cameras can work very well.
In lower light ("gigs and shows", street in low light,...) you need a large sensor - at least APSC.

Do not be mislead by megapixel madness. More is usually better all else being equal, BUT all else is seldom equal and higher mp is often at expense of final quality.
If your budget is limited, you can get very very very good results from 6 megaixels - better than what was considered 'wedding quality' not so long ago!. So an old second hand 6+ mp DSLR can do a very good job in many cases.

You do not want a camera that is easy to understand - you just want a capable camera with a "bunny mode" that you can use until you get used to it. ALL well made cameras are easy to understand after a while - you just need a friendly learning mode.

Lenses that you will require depend on desired style etc. A mid range zoom is a desirable start. A 50mm or 35mm f/1.8 or better prime will be your great friend in due course but is not necessary initially - but if a good & cheap one wanders past, you want it.

Great quality is in very large part dependent on the part of the overall photo taking system that holds & owns the camera (ie = you, in case that wasn't clear :-) ).
"Crisp" is achieved with either a horrendously expensive lens used well in almost any circumstance OR a modest or better lens stopped down, or not used in ultra low light and used very well. That is, a kit lens can often do as good a job as is needed to start. Much better much more expensive lenses can follow along in due course.

To see what the specs of both new cameras and old lower mp models are like in practice look at a quality review site and pore over both samples and tests. I recommend the superb DPReview site , but there are many others.

As an example - DPReview provide
General news
Reviews
Camera database
Lens database
Technical articles
Forums & competitions
and more.

Depending on budget, I'd start by looking at:

If "as cheap as possible" is the aim, reviews etc for 6mp range DSLR's which are available at good prices second hand.

If somewhat more money is available, one of the entry level cameras that has good reviews in the areas that interest you. Note that the DPReview tests and those of all good sites provide sample images taken under a range of conditions. Look especially at the ones that reflect you areas of interest. In my case, I am biased towards cameras that both
perform well in low light conditions
or allow high shutter speeds -

ie gigs and shows, & street at night, motocross, surfing, athletics ... . The two apparently different requirements converge because you need low noise at high ISO for low light photos and you need the ability to use high ISO for action shots so your shutter speeds can be fast. Others are biased towards great dynamic range or awesome colour rendition or ... . .Everything matters, but wht matters most to you needs optimising when you make decisions.

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Added from a comment elsewhere:

@weberc2 mentions Nikon & Canon 50mm f/1.8 lenses. Sony make a similar 50mm f/1.8 lens.
It is featherweight, and uses as much plastic as it can.
BUT optically it offers superb value for money.
It is easy and cheap to make due to the fixed focal length and in Sony's case uses the optical design from a Minolta lens that was introduced decades ago and which has paid its modest development costs many times over.
The entry level 50mm primes from all major makers give you the chance to own a "real" lens at an entry level price.