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I just started playing and am borrowing my friends dice every time we play. I am looking to buy my own. How many sets do I need to buy?

For a player

Three or four sets will cover almost everything you could ever want. This will mean you have at least 2d20’s for advantage/disadvantage, at least 3d6’s to roll for abilities and a handful of other dice for rolling damage or hit points. 4d6 is preferable as it means you can drop the lowest each time.

Different classes have different hit die or some add certain dice to each of their attacks at higher levels. Different weapons and spells also use different types of dice for damage, such as a d4 for daggers. Rolling a d6 six times every time you wanted to attack something would get extremely tedious very quickly. The more dice you have on hand, the faster you can get rolls done which keeps the game flowing.

For a DM

Many sets of dice are extremely useful, some traps, like those found in Xanathar’s Guide to Everything, require 24d10’s! Also, if you run encounters with large hordes of creatures, rolling a fist full of dice is so much easier and quicker than rolling for each creature individually. Having a large stockpile of dice is also nice as it means every player has their own dice to use.

Personally, I have somewhere in the region of 400+ dice, mostly d6’s and d20’s but theres a fair amount of every dice. I simply bought few bags of 100+ dice which weighed about a pound each, my dice-bag bag weighs 1.5kg (which I find to be amusing as 1.5kg is about the same weight as your average medieval longsword).

It is often far cheaper to buy dice in bulk quantities. I often see a bag of seven dice priced at the same amount, if not more, as a bag of 100. I will inform you that these “pound of dice” or “100+ dice” bags primarily consist of factory 2nds. This simply means that, whilst they are perfectly serviceable, they have minor visual imperfections. From my experience, the vast majority of my dice are perfectly fine and I do not see why they would be considered 2nds, you get the exact same types and styles of dice you may find in stores or online. One issue I found when I bought a 100+ bag was that it was hard to make sets, simply because i wanted them to be organised by colour or style. However, this issue was easily overcome by buying 2 more 100+ bags :P

As the others have suggested, I would recommend the standard set of polyhedral dice; a d4, d6, d8, d10, d12, d20 and d100 (although that last one isn't as often used).

I personally also add to this a second d20 (useful for when rolling with advantage or disadvantage) and 3 extra d6s (so that you have a total of 4d6 when rolling up new stats, if your DM uses that method for character creation).

The standard set of 7 polyhedral dice is the bare minimum. It should include dice with four, six, eight, twelve and twenty sides as well as two with ten sides (one will be numbered 1 to 10, the other 10 to 90).

Depending on what class you play you might want to get some extra dice for damage rolls. For example, if you play a wizard or sorcerer you might want to have extra six and eight sided dice for spells such as Fireball and Cone of Cold.

You could also just go on eBay and search for a pound of dice or assorted dice.

I have a single set, including d4, d6, d8, d10, D12, d20, d% and d20. That is the standard for a 'set' and I have never wished for more dice.

That said, some people like multiple sets, and whenever one die rolls badly they switch to another one, so there is lots of room for preference, but 1 is good enough.

Minimally, each player will need a single d20, one die corresponding to your class's hit die size, and one die corresponding to their most commonly used attack. A barbarian with a greataxe can even get away with the d20 and a d12. (Occasionally borrowing a die from someone else for specific situations.)

Of course, usually, you'll have a more complicated character where you have different-sized weapons, abilities which use different dice, and so on. A single basic set of polyhedral dice will cover these, although you may find yourself rolling one of them several times in a row and adding.

Rolling multiple times like that can really slow the game down — not as badly in 5E as in previous editions, but enough that it really can add up over the course of an evening. So, I suggest that you at least have two d20s, so you can roll both at once in situations of advantage or disadvantage.

If you have class features which require an additional die, and you don't have an additional die, that also slows things down. For example, rogues frequently add a d6 to damage — and multiple d6s at higher levels. You should have dice on hand for any frequently-used features like this. But this really varies by class selection and even particular path chosen, so there's no general rule.

When I've run games for middle-school kids generally with no previous D&D experience, I have actually given each kid a set of dice pre-matched to their pre-generated character. Two d20s for everyone, and other dice for their main attacks and signature features.

If you're a spellcaster, at higher levels, you'll probably want a wide selection of dice, because spells often require a handful of them. For example, good old fireball does 6d6, and cone of cold is 8d8. If you do that a lot, rolling eight times and adding as you go will get tedious for everyone. You don't necessarily need all eight, though: rolling 4d6 or 4d8 twice will do.

(Bonus tip: roll your d20 for to-hit and the damage die or dice at the same time too, rather than one after the other. If you miss, just ignore the damage die, and if you hit, it's already rolled.)