You probably can't do much better than that TT/amp combo in that price bracket. As to the speakers, there are many around, so auditioning is well worth it. If you have a hi-fi specialist nearby who carries Rega and a range of comparable speakers, you could ask them whether they will do a demo in your home. That way you'll be able to hear exactly what a set-up will sound like in situ.
Pro-ject TTs are definitely worth considering (the Debut Carbon for example) but the Rega RP3 works so well with the Brio-R that I would consider that your first option. I compared the Brio-R with the Marantz PM6004, and the difference was staggering. The Marantz sounded like a decent amp reproducing a recording - comparable with what I already had - but the Brio made the music sound almost live by comparison, as though the musicians were playing in my living room. I was amazed that spending a bit more could make such a difference.
I'm not very au fait with the various speakers available, but do consider Monitor Audio Silver RX6 speakers. They are what I plumped for with that TT and amp. They are floorstanding speakers, so you will need floor space rather than shelf space for them; the advantage is that you won't need to budget an extra couple of hundred pounds for speaker stands, so all your money goes on the sound-producing equipment. They producing a bag of punch, but still cope with fine details. They are just the acceptable side of too exciting for me - the Brio-R seems to rein them in a bit. I'm currently listening to 17th century viol consort music, and it definitely doesn't sound like rock, so don't worry if you mostly listen to acoustic stuff.
Don't forget you'll also need speaker cable, which can get expensive if you need a lot of metres to hook amp up to speakers.
I'm not sure I'd suggest going any higher than this range for your first set-up. I'm sure you'll be delighted with the results you get, especially if it's your first venture into vinyl and/or hi-fi, and who knows? In a few years you might find that you're really into it, have a good collection of records and have an itch for something better; or you might still be completely happy with what you have. From what I've read it sounds as though you'd have to spend double the amount to get appreciably better results, so it does get very expensive (as no doubt some posters here could testify!).
How do you currently listen to music? You don't mention a CD player, so I'm assuming you don't want one. But do you currently listen to music from your computer? If so, buying a decent DAC will enable you to get the best out of your digital music from your new amp/speaker combo. It's worth thinking about budgeting that in if you do listen to digital.
Good luck, and happy auditioning.