I have a cheap brand bass at the moment, but I am interested in actually learning to play bass properly. I need a bass to cover most genres but I don't know where to start. Any Advice?

I could spend more if necessary
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Stratocaster fan...
You wont find much for that price. Squier VM Jazz(£220-£250), Schecter Stiletto Deluxe(£200-£240), Spector Q4 (£270).

I really don't see any good bass to upgrade to until you reach the £350 price range. At that price range you can get the Schecter Stiletto Custom, Spector Legend.

If you live in Glasgow, I noticed a second hand Ibanez BTB405 for £240 in the "Guitar Store". Didn't play it though, so i'd advise trying it out first.
Squier Vintage Modified Jazz or P, Yamaha... then as said, you won't find anything nice until you're in the £350-450 range.

Welcome to the world of UK based musicians, lube up your ass because we get screwed daily on musical gear prices.
And yet, to me, what is this quintessence of dust? Man delights not me: no, nor woman neither... nor women neither.