This question goes straight to the heart of wine enjoyment. The short answer is to gain a little more knowledge, be a little braver and indulge your own natural curiosity.
Personal taste is so fundamental to one’s perception of wine quality that the first step to drinking better wine is to gain greater insight into your own preferences for different styles. But many fall at the first hurdle simply for fear of saying the wrong thing and appearing foolish. Wine has a jargon that can seem impenetrable until suddenly you realise that it really isn’t. An introductory wine tasting evening (there are lots of options here) would almost certainly make everything a great deal clearer, but for the purposes of this answer, here is a very quick guide.
Any wine can be described in terms of a small number of attributes: dryness / sweetness (anything from bone dry to the sweetness of a cup of tea with ten teaspoons of sugar in it - if in doubt, sweetness is detected on the tip of the tongue), level of acidity (a sourness that provides a crispness that cuts through the wine and is felt on the sides of the tongue), alcohol (from light-bodied to very full-bodied, this is the weight of the wine, with higher alcohol wines feeling noticeably heavier and hotter on the middle of the tongue and in the back of the mouth), tannins (really only relevant for red wines, these can be so soft as to be non-existent or so hard as to taste like a cup of tea poured from a pot that has been left to stew with too many tea leaves for too long) and oak influence (ranging from completely absent, through creamy, cedar or vanilla character all the way through to a green, sappy hardness that many would find unpleasant).
So dryness, acidity, alcohol, tannins and wood influence are the dimensions of the wine, and provide a pretty objective way of describing it. Once you have decided where you are on the spectrum for each of these, you are in a position to start looking for more of the kinds of wine you like. You’ll find lighter, crisper wines from cooler climates (northern and eastern France, north-western Spain, northern Italy, New Zealand and Oregon, for example). If you prefer richer, spicier, full-bodied wines, then warmer climates would work best (such as southern France, the Mediterranean, most parts of Australia, Mendoza in Argentina, and California).
Armed with a little more knowledge of how to describe your own tastes in wine, the next stage is to be brave. Go to a specialist wine shop if there’s one near you and ask for recommendations. If there isn’t one near you, there’s a huge choice of quality merchants to be found online who will be happy to advise and deliver. Most people work in wine because they love it, so they will be friendly, enthusiastic and informative. You may find one of the few that need to get over themselves, but that’s their problem, not yours. Importantly, tell them not only the style that you’re looking for, and maybe the context in which it will be drunk, but also how much you are comfortable spending. This may help them narrow things down considerably. Similarly, if you are in a restaurant, ask the staff for help. If the person you ask doesn’t know, they will probably know someone who does. And again, they will find it easier to help you if you tell them your spending limit.
Be curious. Ask friends if they have drunk anything good recently. Subscribe to a wine publication (online or print versions). Check if a local wine merchant does tastings. Follow wine writers on social media and read wine blogs. Step out of your comfort zone and try styles you don’t usually buy: just because you have a penchant for dry, zippy whites, it doesn’t mean you won’t like an envelopingly rich, intensely sweet dessert wine or a full-bodied, spicy red.
A few words on price: wine duty is fixed, whatever the price of the bottle, at around £2.50 including VAT, as are many other costs such as packaging and transport. So you will most likely get a lot more wine for your money if you spend £12 rather than £6. But this effect becomes much less pronounced the further up you move in price.
Be adventurous, be bold and be prepared to make mistakes. If your quest for better wine doesn’t take you up a few blind alleys, you’re probably not trying hard enough.