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I'm sure you could go down to PM and say exactly what you've said here and get more recommendations than you can handle.

There are lots and lots of cool, challenging and beautiful puzzles out there - and you live near a warehouse full of them!

I assume there's a retail store, too? and the guys who do their videos must work there as well as Allan Stein who founded the company - I'll bet they all love to recommend stuff.

That said, browsing the site is fun once you find your way around and can avoid the things you don't like

Here's my standard "how to get started" rap (a little old, so some items might be unavailable):

I'd say you probably want to check out several different categories of puzzle:

• Disentanglement - This includes the wire puzzles, but also the various metal, wood and rope and wire and string varieties. I'd look at a couple of the harder wire puzzles as well as one or two metal and string or wood and rope puzzles. If you're new to these, ThinkFun put out a "Houdini - Master of Escape" set that's actually like a step-by-step course in disentanglements
  
• Packing/Assembly - Usually wood or plastic, these range from the ever-popular Soma Cube and other "make this shape from these pieces" puzzles to really complex packing problems - often in the form of "put these pieces into the box so the lid will close" or "fit these pieces in the frame" There are also some interesting hybrids like the 4L puzzle from Cubic Dissection or the LUV puzzle from Rombol (a Stewart Coffin design that's pretty tricky)
  
• Interlocking Burrs - My favorite category. Includes the traditional 6-piece Burr and variations by the thousands. I highly recommend the work of Stephan Chomine and Osanori Yamamoto in this area, but there are lots of great designers. Baumegger, Demirhan, Eyckmans, etc. Do a search for Pelikan (a manufacturer) on various puzzle sites. This is a pretty deep category with lots to explore - one subgenre is the TIC puzzle (Turning Interlocking Cube) which BurrTools won't solve (look for work by Ken Irvine, Jeff Namkung or Jos Bergmans). See more below
  
• Puzzle Boxes - Lots of fairly shoddy ones around, but some very nice things out of Japan (check the Karakuri Creation Group). The good ones tend to be expensive.
  
• Twisty - Rubik's Cube and its descendants - not my thing, but there are a million variations out there.
  
• Sequential Movement - sliding block puzzles and various sequencing puzzles (sometimes an overlap with the Twisty or Disentanglement categories). A great, cheap example is Rush Hour from ThinkFun that, like the Houdini set mentioned earlier is like a graded course in sliding block puzzles
  
  Anyway, I'd say try a couple of disentanglement things (both wire and wood), a packing puzzle or two and a few interlocking burrs (I recommend trying a relatively traditional one and a couple of the fancier ones from Pelikan or Cubic Dissection). See what sort of things appeal to you and follow on from there.
  
  Do check out the links in the sidebar, too. Allard's Blog and PuzzleMad are great, Rob's Puzzle Page is vast and you can learn a lot just by browsing PuzzleMaster and some of the other shops.
  
  Here is a list of wooden burr-type puzzles under $30 that I have played with and enjoyed - I made this awhile ago so thy may not all still be available, but this should get you started
  
  **Open Box Packing aka Deadly Romance - a really nice caged burr
  
  Cross Cage by Tom Jolly - a burr? 3D maze? Pretty cool
  
  Four Caged - a really tough little caged burr
  
  Epsilon - a knockoff of Vertex Burr #1 , originally designed by Yavuz Demirhan in 2012 - great puzzle
  
  Shape Shifter - very tough assembly puzzle - soma cube on steroids
  
  Sarcophagus - surprisingly challenging. I have a thing for "3 sticks in a box" puzzles like Tribord
  
  Matchbox aka Oscar's Matchboxes- tricky and a cool idea
  
  Double Saturn - similar to some of Osanori Yamamoto's classic work, this is tough little Pelikan-style puzzle on the cheap
  
  Four L - like The Double Saturn above
  

• NOT to be confused with the **4L from Cubic Dissection (which is a truly great puzzle,but OOP now)
  
  Quadstair
  
  Four In The Box - just above the $30 limit, but it's pretty fun and two puzzles in one
  
  Feel free to ask questions on the sub about specific puzzles or categories
  
  Puzzlers are a pretty supportive community - welcome!

I only want to add a few things to this excellent answer: This book by Steward Coffin is another one I like to recommend. There is also this one you can read online for free.
I do also recommend starting out with a simple burr puzzles, for example with one of the classical 6 piece burr puzzles. If you find good square stock you can get a long way with a good hand saw and a chisel - but if you have access to a table saw even better:)

Then also consider the difficulty: There is the level (a sequence of numbers that tells you how many moves you need to remove the n-th piece) as well as the fraction of valid assemblies / total number of assemblies, that tells you how many configuration of pieces that could theoretically form the desired shape are actually physically possible. In the case you're new to woodworking I recommend starting out with a puzzle that is easy to manufacture. The ones without any internal edges and corners are sometimes labelled as "notchable", which means that they can be made using only straight cuts. (There is also "millable" which relaxes this condition to allow for internal edges.)

Lastly I recommend the "burrtools" software, as you probably need to try to assemble the puzzle a few times to check for tolerances etc: It is made for computing solutions to burr puzzles - but it takes a while to understand how to use (there are great instructions though) - if you need any help with this or just want a finished file for a specific puzzle just ask!

And finally: Don't forget to post your progress/result here :)

Not sure if you already have gotten this one but this one is fun.

Here's a really cheap addon puzzle that you could throw with any other gift.

I would really recommend this though. It's easier to solve than a Rubik's Cube, and to someone not familiar with all the shapes they come in, it might be novel.

Search for Hanayama on Amazon. They are rated level 1-6. Skip levels 1-3 and start at level 4 (which are still quite easy). Level 5 and 6 are the best ones. For me, Baroq is the most fun (but easy); Enigma is easily the hardest while still being unique and fun.

4L from Cubic Dissection is a great one! My mother has had my copy for 6 months now and still hasn't solved it.
Half Lid Box from them is also a great puzzle.
Another packing puzzle I really love is Redstone Box
https://www.amazon.com/Creative-Crafthouse-Redstone-puzzle-teaser/dp/B006IGBPPY?ref_=bl_dp_s_web_7835094011

MARBLE Hanayama Cast Metal Brain Teaser Puzzle (Level 4) https://www.amazon.com/dp/B0053163VG/ref=cm_sw_r_cp_apa_c4QtybZST9V3N


This one's pretty easy and very fun to just play with... it like screws together when you line up the lines

This puzzle box line is okay. Of the 4 I've seen, this is my favorite:
https://smile.amazon.com/dp/B072NBSTDM?psc=1

I like both of these trick locks, too.

https://smile.amazon.com/Houdini-Under-Lock-Puzzle-Teaser/dp/B00CFDMFWQ/ref=sr_1_2?s=toys-and-games&ie=UTF8&qid=1498260348&sr=1

https://smile.amazon.com/Recent-Toys-Houdini-Lockout/dp/B00QEEWLSU/ref=sr_1_17?s=toys-and-games&ie=UTF8&qid=1498260348&sr=1

Thanks for posting this (and Nutcase)... In for one of these as well!

Labyrinth is the same price too!

https://www.amazon.com/gp/product/B0038BCMYU/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1

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Not sure why they don't come up that price during a normal Amazon search though... had to use your links for some reason

Looks similar to this one, especially the later preview images https://www.amazon.com/KINGOU-Wooden-Fifteen-Oblique-Puzzles/dp/B00S87D6US

There appear to be several possible ways to combine the sticks. You might find a solution here:

https://www.google.com/search?client=safari&hl=en-us&q=interlocking+Sticks+Puzzle+solution&spell=1&sa=X&ved=0ahUKEwjW7r3JoL7XAhWLw4MKHSIgBqsQvwUIISgA&biw=768&bih=928

Imagine this is along the lines of what your lookin for?

https://www.amazon.com/Wooden-Puzzle-Gift-Set-Box/dp/B005JEE3IW

It's called Double Disk. https://www.amazon.com/HOGWILD-Double-Disk-Puzzle/dp/B00630H270

You're missing a piece. There's supposed to be a oval shaped loop at the top that is to be removed. Similar to this: https://www.amazon.com/Go-Games-Puzzlers-Vertigo-Puzzle/dp/B00NGRCYIU

Not the greatest image but I couldn't find a better one.

I'm embarrassed to say I used the wrong name, this is the CLAW puzzle not the CRAB puzzle (which doesn't exist).

You can buy it (and see it) on Amazon or in a variety of stores, I know these puzzles are sold at Target and Wal-Mart (among other retailers).