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

Amazon is selling the KEF X300A (which includes  a DAC + 2 amps for each speaker) for $599.98. It is pretty good for the desktop. I have had the KEF LS50 on the same desktop and that was definitely better but the cost of that system was close to $5000.

I bought the X300A and I am quite happy with the sound, especially for the price I paid.

BTW - There will be a new KEF ACTIVE LS50 available in Dec 2016 for $2200

I agree with the idea of going for active, or at least powered, speakers. Quality of sound for the money is very good when spending less than $1,000, especially when spending way less than $1,000. Then all that is required is an input (laptop, phone, ipad, etc.). Simple, convenient, good sound, portable.

For a 21yo girl I'd get the Audioengine HD6 and be done with it.  Includes built-in class A/B amps, DAC, Bluetooth receiver, analog input/output, and a remote.  Finishes are beautiful as opposed to most other competitors in this category, which I'd think she'd also appreciate.  All she needs is her phone and she's done.  I bought the B2 in walnut for my brother.  It looks and sounds great and he loves it, and the Bluetooth reception is excellent so I'd expect similar from HD6.  BTW, Dagogo reviewed them and greatly preferred them in direct comparison to Vanatoo.  You can read multiple reviews on their website.  30-day trial period as well.  Best of luck.