Lots of elements….
I’d agree with the comments that suggest selling the pre-amp.
Is there a cartridge/ needle on the tone arm on the turntable? If yes, can you see a name/ model?
You should be able to get sound from the streamer once you have cables and speakers - just try a free trial on Tidal perhaps?
The Pink Triangle will need a cartridge and a suitable phono stage and a cable from turntable to phono stage and another from there to Supernait. Please don’t spend much on cables.
A fairly modest (say) Ortofon cartridge will do the job if you need one. If in doubt. Take it to a shop and have them fit a new cartridge. As others have said, a Naim Stageline or Rega Fono may well suit as a phono-stage and should be a safe second-hand purchase.
Naim standard cables may well be cheap second-hand from eBay or Pinkfish, and are good choices. Richard’s link should pin down what you need.
Whatever speaker cable you get should be at least 3.5m long and the same length both sides. Naim A5 is annoyingly inflexible but works well, and again eBay may be a help.
Speakers really don’t sound the same, so it is hard to be definite without knowing which one you like/ dislike، how the room sounds and so on.
You can buy a switch box that sends signal from speaker cables into speaker cable A, speaker cable B or both. QED charges a bit over £100 for a decent new one - it’s not for purists but works pretty well.
Speakers often sound better if not in a corner and some want some distance from a wall behind them and some toe-in ( turned a bit towards you) too - you may need to experiment.
Some bookshelf speakers really mean it, but most speakers that don’t go on the floor (with or without spikes) need stands to work properly. EBay again?
EBay suggests you could probably get decent speakers from Neat, B&W, Spendor, Castle, Ruark, PMC, Linn and others for under £300. They don’t sound the same, and will suit some places/ ears better than others, but there haven’t been many poor speakers from any of those brands in a long time.
If you don’t know what speakers would suit, it may be safest to hear a few at dealers. If that means buying new, B&W 607s and Wharfedale Diamonds will be among several options worth considering - and in-budget, if ex-dem.
Alternatively, you can risk splashing out £200 or so on good s/h speakers, perhaps after (a) hearing a few different speakers to get some idea what you like and (b) bearing that info in mind when reading a few online reviews. If you buy and then really don’t like the sound in your room, you too can always sell them.
You have to see speakers as well as listen to them, so I suggest looking at some pics on eBay or ‘best speakers’ articles to see what sort of thing you would like/ hate to see in the room.