You haven't yet specified a total budget. So unless you're willing to spend quite a bit more then at this stage my amp and speaker recommendations stand. You can't really go wrong with Cambridge Audio amps. The only one that I would avoid is the very basic AM1 (£79) simply because it doesn't have tone controls and I think there might be times when they come in useful with vinyl on an entry level system.
The other bits you will need are some cables to link the turntable to the amp, and then some speaker wire. The turntable connection leads are the phono to phono cables I mentioned on the previous post. The free ones supplied with some bits of audio gear are pretty crap. You're looking for something shielded as a better choice. In the absence of any real solid technical info to the contrary then
these are probably just as good as anything a Hi-Fi dealer might try to flog you for £10-£25.
Speaker wire, a budget all-copper cable of a cross sectional area 1.5 square millimetres is adequate so long as you're not running more than 10 metres per side. Look for a 42 strand cable rather than something with hundreds of strands. Hundreds of very thin stands makes the sound worse rather than better. 42 strand is optimum for cables between 1mm and 1.5mm cross section. 79 strand is best for cables at 2.5mm cross section area.
Here's something very good value in that size.
The above covers the basics which will get you going. There are some extras that are very useful, and other things that make life easier or keep things neat.
The first is a record cleaning brush. Personally this is something that as a vinyl user myself I'd class as an essential. When the needle travels over a dusty record some of that dust gets embedded in to the grooves. That's what you hear as crackle. Using a brush won't remove crackle from dust already embedded, but it will help slow down the build up of new dust and so keep your records sounding better for longer. The design I have been using for 30+ years has a felt pad flanked either side by carbon fibre bristles. Google "Super Exstatic Carbon Fibre Vinyl Record Cleaning Brush". The same design is sold in Hi-Fi stores for £24+
The second thing is banana plugs. These are ends for bare speaker cable that make plugging in easier. You don't 'need' then as the speakers and amps all have connecting posts that will take bare wire. It's just a bit of a faff, especially if you're not used to wiring up speaker cable connections. Ebay is fine for these things. You don't need that many, four for that cables at the amp end and four for the cables at the speakers ends; so that's 8 in total.
Good luck