The acrylic paint set with sixty colors for twenty dollars and the one with twelve colors for sixty are not the same product with different amounts in the box. They are made differently, they behave differently on the canvas, and the cheaper one will cost you more evenings.
Here is what the differences actually are, and what a beginner genuinely needs.
What acrylic paint is

Pigment particles suspended in an acrylic polymer emulsion, thinned with water. As the water evaporates the polymer particles knit together into a continuous flexible film with the pigment locked inside it. That film is water-resistant once formed, which is why acrylic cannot be reactivated the way watercolor can, and why a brush left to dry is usually finished.
Everything that varies between paints comes down to what is in the mixture besides pigment.
The four things that separate a good set from a poor one
1. Pigment load
The single biggest difference. Artist-grade paint carries a high proportion of pigment; student and craft paints carry less pigment and more filler and extender.
You notice it immediately in three ways. Weak paint needs two or three coats where good paint needs one. It shifts noticeably as it dries, usually darker and duller, so what you mixed is not what you get. And it goes chalky when mixed with white rather than staying clean.
2. The word "hue"
If a tube says Cadmium Red Hue rather than Cadmium Red, it does not contain cadmium. It is a mixture of cheaper pigments approximating the color. That is not automatically bad, and some substitutes are excellent and less toxic, but a hue behaves differently in mixtures, usually going muddier faster.
It is worth knowing what you are buying, because the word is easy to miss on a label.
3. Lightfastness
Printed on any tube worth buying, usually as an ASTM rating. I is excellent and II is very good. Anything rated III or below will visibly fade, and a painting hung on a wall for five years is exactly the situation where that shows.
Sets that do not print a lightfastness rating anywhere have generally not earned one.
4. Body
Three broad consistencies, and they are not interchangeable.
Heavy body is thick and buttery, holds the mark of the brush or knife, and is what you want for texture and impasto. Soft body or fluid flows readily, levels out smoothly, and suits detail, thin coats and even coverage. Craft acrylic, sold in squeeze bottles, is thinner still and formulated for decorative work on wood, terracotta and similar. Generally lower pigment, generally not lightfast, and perfectly good for what it is for.
Most tubes also mark opacity with a small filled, half-filled or open square. Opaque colors cover; transparent ones are for glazing. If you want to cover something, check the square.
What a beginner actually needs
A small set of good paint beats a large set of poor paint, and it is not close. Eight well-chosen colors will mix almost anything, and they will mix clean.
A working starter palette looks like this: titanium white; a warm and a cool version of each primary, so a slightly orange yellow and a slightly green one, a slightly orange red and a slightly violet one, a warm blue and a cool blue; and an earth color such as burnt umber.
Two colors of each primary is the part people skip and it is the part that matters, because a clean green needs a cool yellow and a cool blue, while a clean orange needs the warm versions of both. Try to mix a bright green from a warm yellow and a warm blue and you get mud, and no amount of technique fixes it. The color wheel explains why.
Black is optional and often best left out at the start. Darkening a color with black kills it; darkening it with a little of its complement keeps it alive.
Why the sixty-color set is a trap
Open one and most of what you find are tertiary colors you could have mixed in a moment, in tubes too small to finish a painting with, at a pigment load that means you will need three coats of each.
In practice people use six or eight of them and the rest go hard in a drawer. If a large set is what your budget allows, that is a reasonable choice, but buy it knowing it is a set for experimenting rather than for finishing work you intend to keep.
Keeping a set alive
Acrylic's speed is its virtue and its problem. Wipe the thread of every tube or pot before you close it, because dried paint on a thread eventually welds the cap on. Work from a stay-wet palette, or improvise one with damp kitchen paper under greaseproof paper, if you mix more than a little at a time. And clean brushes as you go rather than at the end, which our guide to brush types and the layering piece both come back to, because acrylic in a ferrule is permanent.
On thin coats and drying times, layering acrylic paint covers the timing.
A paint by numbers kit is not a paint set
Worth saying plainly, because the two get compared as though they were competing purchases.
The pots in one of our kits are mixed to one specific design and supplied in the quantities that design needs. They are not a general-purpose palette, there is no white to extend them with, and they are not intended to be mixed. That is a feature of the format rather than a shortcoming: not having to mix is precisely why a numbered canvas settles people the way it does.
So the two answer different questions. If you want to make your own decisions about color, buy a proper set of eight good tubes. If you want a guided evening with the decisions already made, buy a kit. Plenty of people do both, and a good many arrive at the first through the second.
If a color runs low mid-canvas, there is a matched extra paint set for every design, and the full range is here.