I mean… some dork wrote about a bunch of puzzle games here (there are even links to all the puzzle games talked about on ye olde sb in there):
I would say that my faves are along the lines of Hanano Puzzle/Jelly no Puzzle, various Picross variants, Toki Tori 2, the Hexcells trilogy, DROD, and Corrypt. Probably The Witness when it isn’t annoying me. The thing is that the harder ones appeal to me, so they might not be the best ones for a beginner.
So I’ll mention some of the easier (or at least easier to get into) ones that I dig, then I’ll name some that might be harder but represent different kinds of puzzle games.
Puzzle games that aren’t so hard
Toki Tori 2+: It is a kind of platformer/metrovania puzzle game where you play as a flightless bird who must interact with the various animals in an area in order to proceed. It gets very hard if you want to be a completionist, but simply getting to the credits isn’t that difficult. Getting all the trinkets is a legit challenge but they are all optional and ideally the main run of the game will teach you enough along the way.
Hitman GO (and perhaps the Lara Croft/Deus Ex ones: Generally very cheap if not free and start off simple enough. Hitman GO did get trickier later on but you generally only have a few moves available at a given time.
Corrypt: Little freeware puzzler with a great gimmick I shudder to spoil. It starts off as a “push blocks around so you can exit the screen” kind of game that quickly mutates into something else entirely. It is relatively short and many will fail to beat it their first go around, so no real pressure on that front. Also the gimmick is among the most brilliant things I’ve seen in gaming and allows you to bend the rules of the game a great deal, so you won’t get stuck trying to figure out the one true solution.
Mole Mania (maybe Donkey Kong '94): Nintendo made gameboy puzzlers (DK '94 is a puzzle platformer which is slightly different). They do get tricky later on but Nintendo isn’t going to toss you into the deep end right away.
Portal: I figure this is fairly well known and needs no intro, but the original in particular is a very clever game, doesn’t overstay its welcome and isn’t that difficult to start with.
Different kinds of quality puzzle games
Hanano Puzzle/Jelly no Puzzle: These are your prototypical “pushing blocks on a single screen to achieve a goal” games. They are devilishly hard and don’t believe in things like having the first few puzzles be easy, so might be rough on a beginner. They might also be the best designed puzzles in this sub-genre.
Picross (and Hexcells): I have no clue if you are the type who would like number-based puzzle games. If you do then there are several good options. Picross is one of the classic ones and if you have a DS then Nintendo made both a high quality traditional and 3D version of it. On the PC there is the Hexcells series, which is basically a clever hybrid of Picross and Minesweeper. The first Hexcells game would be best for a beginner as it takes the most time to explain its rules.
Spacechem: Absurdly fucking hard. It is basically a programming-type puzzle game. You are given a grid of squares and must place symbols that basically create a program that take a given set of inputs and produces the proper output. What it does have in its favor is a demo that is very substantial, I believe giving you the first two worlds to play through. That would give you both the whole intro section plus enough actual puzzles to determine if it is something you might care for.
DROD (likely Gunthro’s Epic Blunder): I did a whole let’s play for the first one of these last year. It is basically a puzzle game in action/adventure game clothing where enemies only take one move after you take one, meaning you have to figure out how to take advantage of this to clear a room of enemies without dying. The fourth game, Gunthro’s Epic Blunder, was made to be an easier entry point to the series, although half the point of the series is how hard it can be.
The Talos Principle: A first person puzzle game by the folks behind Serious Sam. Sounds like a bad idea, turned out pretty good. I thought it ramped up a bit slowly in terms of difficulty, but that might be a benefit to you. Also comes with a pretty good framing device examining the meaning of existence and such.
The Witness: Another first person puzzler, but it takes place on its own Myst-type island and has a unique mix between traditional “puzzles on a screen” and “puzzles involving the environment” that can sometimes overlap.
I… should probably stop here. If any of those intrigue you mention it I can can probably either talk more about it or recommend other games in that vein.
Also, two tips that are likely obvious but often help me out in these kinds of games.
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Often times you only have so many legit moves to start with, particularly in games that separate moves into distinct actions (i.e. you push a block one square). Learning to recognize them and which ones are the most promising is key. Better puzzle games will take advantage of this at some point to make the least promising one the right one, but generally not until later on.
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Figure out what the final move or solution has to look like and work a few moves back from that. Often times beyond the final goal a few elements will have to be placed in a certain way for it to be possible, and identifying them early on can be a great help.