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

Note: There are a bunch of great Pelikan puzzles out since I originally wrote this: Tom Pouce, Triad, Wing Hanger, Petit Pack, Little Hug, Angelus, Addition, Crystal Ring.....and that's just some.

Feel free to ask questions on the sub about specific puzzles or categories

Puzzlers are a pretty supportive community - welcome!

I remember playing the physical version of this as a kid, was pretty fun actually: https://www.amazon.com/ThinkFun-Rush-Traffic-Logic-Girls/dp/B00000DMER

With ThinkFun, were you looking at games like this? I played games like that in the gifted classes I had in elementary school, and they were a lot of fun. These days, you can even download that type of game for an iPhone or iPad for 99 cents, or for free.

For older kids (say, 10 and up), the card game 24 is great for building critical thinking and math skills.