Hifi is the persuit of perfect sound quality. For the scientists amongst us, this is the perfection of absolute flatness ie totally faithful reproduction. Colouration is bad and is generally looked down upon as a bad trait. "Hifi" is usually the preserve of the rich(er) enthusiast, technical people and/or those who get carried away with brands, 'what hifi'-style language and perceptual differences based on the amount of money they have spent rather than what they actually here. In reality, for most people it is a way of tailoring the various distortions to suit their tastes.
Studio generally tends to mean "As cheap as physically possible whilst doing the job". What "the job" is depends on the studio! Abbey road use B&W Model Nautilus speakers, others use huge PMC monitors while some use crap powered monitors bought with the change from the upgraded desk simply because it had more lights.
In terms of ampification, the term "studio" or "PA" should just be avoided. For the home, even if it is a cheap one, get a good HiFi stereo amp.
On the second hand market with an absolute limit of £500, you can get some reasonable stuff, extending this to flex to £800ish ought to get you some very nice kit indeed if you are prepared to do the legwork beforehand to know what you like and then wait for it to come up on the S/H market.
If you are going to go down the S/H route with that kind of money, assuming you will be using the phono stage found inside the amplifier, I would be looking for about a 50/50 split, perhaps more on the speakers if it means you get some in better condition (as they are the most easily damaged). For the legwork, I would be in hifi shops listening to new speakers in the £600-£1200 range (depending on how old you are prepared to go with s/h speakers) and integrateds in the £500-£1000 range.