I found my kid more interested in shapes than numbers, adding 14 and 7 to get 21 is boring, while triangle, rectangle and heart shape is fun.
But what could I introduce, besides the basic shapes, e.g. triangle, rectangle, parallelogram, diamond, circle, line, and angle?
I couldn't really go on to Elements; that will kill the interest. Any suggestion on how to keep the introduction basic and fun and still encourage some thinking?
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If it were me, (assuming a child in elementary school) I would start heading into drawing with a ruler and compass.
Working on accurately measuring lengths and angles, drawing circles, lines, and perhaps other shapes, depending on the tools you add, would introduce a visual-art aspect that might help fuel further interest.
For instance, you could work on constructing an equilateral triangle, then adding a few iterations to make it a Sierpinski triangle.
Or you could construct the various centers on a triangle and show they're collinear.
Or you can work on drawing some simple fractal patterns.
Color and pattern goes hand in hand with art and mathematics.
Origami is also a good way to go, and is a good option for children who can't be trusted with sharp implements. There's a book called "Geometric exercises in paper folding" by T. Sundara Row that was passable, but I imagine there's newer and better books like that out now.
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Puzzles are an excellent way in invoking interest in math. I would look up geometry puzzles online and give him/her some to work out. Increase the difficulty of the puzzles as your kid gets more of an understanding of geometry and along the way, you can introduce geometric concepts that would help in solving the puzzles. And because those concepts would've helped him/her in solving the puzzles, they'll be more likely to be remembered. Better to teach your kid the concepts yourself through problem solving than to have some teacher in the future try to just cram seemingly irrelevant junk into their mind. That's just my theory though.
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I highly recommend the Keys to Geometry workbooks, which are how I fell in love with geometry and construction myself as a kid.
They require no preparation at all. If your kid can draw a straight line with a ruler (straightedge), he/she is ready.
The books are almost entirely interactive. Get a good compass, and a straightedge. (A ruler is okay only if your kid understands not to use the markings on it at all.) By book seven your kid will be able to construct a perfect regular pentagon on a blank sheet of paper, unassisted.
I have also supervised several students through them (graded their exercises) and I recommend reading the material in the teacher's manual, if possible. (Or you can just get the workbooks; you'll be able to check the work fairly easily.) The teacher's manual contains a section on how to instruct beginners to use a straightedge correctly and precisely, how to hold a compass, etc.—which is likely easy for you to do but is more difficult to put in words so that a child can actually learn it based on your description.
I will also note that it is important that the work be checked for precision. Don't tolerate sloppiness even a little bit, because if the ability to do the instruction "Draw an arc with center X" is missed (slightly off from X, or a slightly imperfect arc) in book 1, then the later books will be impossible. These books have extremely little repetition. That's one of the reasons I loved them so much.
These books very definitely encourage thinking.
I've seen teachers of higher math criticize them because they don't contain actual proofs. Instead, they have the student work through some concrete examples ("draw a triangle"; "construct the perpendicular bisector of each side"; "do they meet in one point?") and after working through several such examples, ask the student to choose the right word: "The perpendicular bisectors of the sides of a triangle (always/sometimes/never) meet at one point." All basic rules are demonstrated in such a fashion.
"That's not a proof!" some say. But this is missing the forest for the trees; other such examples where the answer is "sometimes" are conclusively proven by example, and having the student find the answer for himself or herself is far, far better than reciting a litany of "rules" and then claiming this "proves" something.
(Newsflash for higher math teachers: the only purpose for a proof is to make someone realize something true. If you convince only yourself but not the student, you have failed. Okay, off my soapbox.)
Point being, this is an accessible approach to the elements, that kids as young as 8 can do. And high school students benefit as well. (Most modern geometry texts have far more text telling the student what is true, and very little requirement that the student work it out himself or herself with an actual compass.)
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I know you asked about geometry specifically, but it is not the content but the logical reasoning that counts. Find puzzles and recreational math/puzzle books! Puzzles can include tangrams and metal puzzles. Books might have an assortment of interesting stuff like symmetries (you can play with mirrors), tessellations, the Mobius strip and variants, spirographs, fractals, packing problems, and more. Most of these can be fun for little kids and there is no real need to bother with so much 'mathematics' at this stage. Just playing with those stuff is good enough!
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Some other ideas:
Children at that age have often problems with perspective. Teach him how to draw using lines that go out from the same radiant. Show him how to draw parallelogram using that technique.
Show him (and/or teach him how to draw) some optical illusions like impossible triangle or elephant with 5 legs.
Show him how to construct a kaleidoscope using two mirrors. Later it will be easier for him to understand concept of central symmetry.
Buy him and/or show him how to make on his own some folding books.
Show him how to make basic polyhedrons or other shapes using stiff wires.
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It's Not Too Soon to Inspire Geometric Thinking
As they manipulate shapes in daily play and become increasingly aware of spatial relationships throughout their day, preschoolers are actively developing early concepts of geometry. At the preschool level, children learn to recognize geometric shapes by viewing and identifying them as a whole object. Providing children with opportunities to explore and experiment with shapes and their properties allows them to move to the next stage of geometric thinking: understanding the individual characteristics of each shape (for example, a triangle has three sides).
Make Geometric Mastery a Game
Here are some easy ideas on turning playtime into math time!
Block play is great for shape recognition
For example, the simple process of matching and sorting similar blocks at clean up time builds geometric thinking. During block play, pose open-ended questions that spur and advance your child’s geometric thinking. Engage your child with questions like:
How is your block tower different from mine? What will happen if I take out the bottom block? Can you tell me what I need to do to make my block building look just like yours? Can you put 2 blocks together and make another shape? We are getting ready to clean up now - how will you remember what you built? Make spatial vocabulary a regular part of conversational language Acting out stories such as "The Three Billy Goats Gruff," for example, offers numerous opportunities for children to show their understanding of spatial concepts like: under, over, across, near, far.
Turn geometric awareness into a game
Increasing your child's awareness of shapes all around him is as easy as incorporating shape conversations into your playtime. Keep shape blocks or cardboard or paper shapes on hand, and ask questions like:
Have you seen this shape before? Where have you seen it? Can you find a shape like this in our home? Do you think this shape would roll or slide? Can we stack these shapes? Can you cut this paper to make another shape? Can you make a square (circle/triangle) with pipe cleaners/yarn
Resources:
Copley, J. V. (2000). The Young Child and Mathematics. Washington, DC: National
Association for the Education of Young Children.
van Hiele, P. M. (1986). Structure and Insight: A theory of mathematics education. Orlando, FL: Academic.
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While taking him out for a walk let him see shapes in buildings, parks, nature etc. Encourage him to draw at first what he saw after his verbal description of geometrical figures, to the extent he can. You should yourself never draw, but only correct/refine his drawings and and related narrations.
I've assisted in teaching kids aged 10–14 mathematical art, and found a lot of success with the following project:
Invent a polyhedron and try to make it! Basically, take construction paper, and have them use straightedge and compass to make shapes and subsequently cut them out. The most "appealing" are regular shapes. Since your child is 4, you should probably cut the shapes for them.
Here were some especially nice moments:
One student tries to make a vertex with $6$ equilateral triangles and asks me why it wasn't working.
Multiple students worked on constructing pentagons, which was a bit of a challenge (for me too!)
Some students finished early and started coloring their polyhedra, with $4$ or $5$ colors. Of course I bothered them by asking if they could have done it with fewer.
Some students couldn't "See" the shape, so we would use projections to form planar graphs, or use "nets" to help. This was often a bit of a brain cramp but a lot of fun.
here is a blog post chronicling a collection of activities we did (some were more advanced than others: triangle inequalities, pythagorean theorem, finding a way to approximate pi with arbitrary precision etc.) that might have some nice ideas for you!