Recos for entry level Bookshelf speaker $400-750 for 21yo nieces first system?

Peachtree Audio "deepblue2" Bluetooth speaker with 440watts of Class D amplification built in,stereo tweeters,stereo mid range drivers & a 6" bass driver in a sealed enclosure 9"high x 14"wide x 6.5"deep @16lbs..Automatic Digital EQ keeps tone naturally balanced regardless of room acoustics.Optical Digital Input & 3.5mm Input included...
I have owned PTA gear & they know how to make music so the sound should be very good.

REGA R1 / RS1. A New Budget Reference
http://stereotimes.com/speak121305.shtml

"... The Rega R1 becomes my new budget reference speaker. In addition of its ability to get the fundamentals of music right, it adds clarity and resolution, and an ability to lay out a vivid and coherent 3-dimensional stereo image. In small room applications, what more could you want?..."

Read the whole review.

They are designed to be placed against a back wall ( they even have their own bespoke wall brackets) .... I had them in my office driven by NAD electronics and I currently still use them as surrounds in my HT mancave.

Highly recommended ......they are relatively easy to find new old stock or used (current RX1 model is past your budget) . Used R1 or RS1 kit comes up frequently including on AGON or canuckaudiomart.com or hifishark.com.

Great sensitivity ~ 90db efficiency...easy to drive with smaller amp.

Height...320mm
Depth ...236 mm
Width....150mm
Weight ... 3.5 kg

Polk RTiA1 would be a nice entry level choice.  Small, easy to drive, and the powerport allows for placement close to/on wall.

Can be had new or via Polk's ebay direct site refurbished (with warranty) for less.

Great customer service.

Feel like building them? I have great plans for a pair that will get her into all sorts of music.

I agree completely with freediver's recommendation of the Peachtree Audio deepblue2. I bought one for my daughter to replace the first system I got her, a Pioneer Elite amp, DVD/CD player and Usher 520 speakers. She and her husband wanted a simpler setup because their toddler was beginning to twist knobs and bounce off speaker stands and they, like your niece, were listening mostly to music from their iphones and ipads.

The sound of the deepblue2 is remarkable, I haven't heard anything better at anywhere near the price. The engineering and ergonomics are great and the simplicity and versatility sound like a perfect match for someone like your niece. For a young person who might be changing locations for a few years the ability to pick it up and move it so easily is a real advantage, and its size makes it easy to fit in almost any living space.

I was so impressed with it that I recommended it to some friends who seldom used their conventional system because it was stashed away in a cabinet and, like many other people, the easy access to music on a phone or tablet meant they simply weren't playing CDs anymore. Best part is, the deepblue2 sounds much better than their old system.

My advice would be to spend the same amount of money you were planning on but spend it on a deepblue2 and an iPad. She'll thank you.