So in effect you are looking for a complete system, incl. streaming. And your set-up is close to a near-field set-up. But presumably it will be a listening room, not a desktop or AV-setup.
Lots of good advice in the thread. To add my thoughts:
You may indeed want to avoid getting sucked too deep in the Hifi-world. I've only dipped in with cheaper stuff (amps, docs and speakers), and my takeaway is that there really are no extreme differences and you may be very happy with reasonably priced kit, as long as you don't start thinking about 'what if'.
You should indeed seriously consider separating the streaming from the amp (the day can be in either). It takes consistent effort to make a good streaming app, and many amp manufacturers don't finish it or don't keep upgrading. So one option is to invest partly in Bluesound, or a Raspberry Pi. You could also consider something like the Quad Vena Play if you use it primarily as an amp which also happens to do streaming fairly well (according to a different thread here); then you may in the future upgrade the amp and invest in a separate streamer if necessary.
It is easier and cheaper, though to have a DAC integrated either in the amp or in the streamer.
For the budget I'd think you might divide it either to 750-750 for speakers versus amp/streamer, or 500-500-500 for speakers, amp and streamer. That suffices for a modest system that actually would not leave much to desire (except stands?).
Starting from the viewpoint that most differences are slight and you should look to be satisfied instead of trying for the 'best', here are my experiences/recommendations. Note: I mainly listen to classical, but also to many other kinds of music. I'm not interested in thundering bass, but do like hearing clear bass notes.
For speakers I'd second the suggest of Amphion; I've got an earlier set (Ion) which do well even at low volume and are non-fatiguing. I also have Harbeth P3ESR but actually the difference during normal listening is slight, the Harbeths are better but I don't miss the Harbeth particularly when I've got the Amphions set up. Personally I found the bass with these speakers sufficient for me, but I haven't tried any of the speakers that others call 'exciting'. I also don't really follow why Harbeths don't do rock or pop.
A different option is to have sub. I've used a fairly cheap Cambridge Minx 22 speakers and Minx sub, which give surprisingly good results and are within budget.
For an amp I'd advice one with sufficient power, but not massive amounts. I've found that a lack of power ultimately is noticeable, even when listening at lower levels. It's like the sound lacks a bit of lustre even though you can identify the details (like faded picture). But in your room some 30 W would probably suffice: the amp depends of course on the impedance of the speakers (so check that when looking at the recommendations). Something like a Quad Vena should work, or a Rega Io/Brio; if you go second hand there are very good deals to be found. I'd be careful when looking at cheaper class D amps which sometimes do not have the capability of providing larger peak power.
So don't worry too much about the 'best' kit, but do look at a few indicators like whether the streaming software will be supported in the future and is user-friendly, what the impedance is of the speakers and whether the amp can provide sufficient (RMS and peak) power into those speakers.
But surely the real experts on the board can fill you in better than I can.