I have always struggled with 3-D shapes; meanwhile, my dad was a Tetris addict. Hopefully by adding some fun games that require visualizing and playing with shapes you can help your kids become more visually "able".
Here are a few ways to do that:
Set - Although this is a card game and it is more about pattern recognition, I think it can be added into this category. It is a bit difficult to explain the rules but it requires you to pay attention to shape, pattern and number simultaneously. There is even a free online game each day that allows you to play against the computer.
Paper Sloyd - I think the easiest way to explain this is "useful origami". In some schools it was used in the early elementary grades to prepare children for geometry. Paper folding gives them the physical experience of shapes - it becomes obvious that a triangle is half a square, etc. The link is to a whole booklet of these types of activities.
Origami - There are tons of online tutorials and books about this traditional Japanese folding art. This is the book I used growing up and still have - it is simple and in color - but also out of print. There is even Star Wars origami (it is actually pretty complicated). This past week I was with some 8 and 9 year old boys. I was shocked at how many had never folded a paper airplane before. We had a blast doing it! Origami can be pretty complicated so make sure you check the book. Some kids may never progress past the beginning stages. That's fine - it has still helped them to better understand how shapes work. I don't encourage you to be over bearing about the connections - just make observations as you go along. You will be amazed at how much they "see" on their own.
Tangrams - This old Chinese puzzle uses 7 shapes to create a variety of different figures. You can get a printable version for free (print on cardstock, laminate them or use contact paper if you have it). You can search for tangram puzzles (here is a starter location) and find a zillion. Typically the figure is solid black and you have to figure out how to put the shapes together. The solution has the pieces slightly spaced apart so that you can see how it was done. There is even a full book of puzzles. We have an older version of this game where you race to see who can solve the puzzle faster (and you get 2 sets of tangrams and a bunch of puzzle cards).
If you are looking for a more typical "curriculum" to teach geometry (for whatever reason) you might want to try Math Mammoth's series. Here are the resources for grades 1- 3, 4 - 5 (with videos) 6th and 7th (book 2 and 3). If you want a collection of worksheets for more practice for 5th - 8th grade check this set out. Although not a full curriculum this is a fun place to play with geometric ideas.
Mapping - Physical maps are actually quite abstract but you can begin by "mapping" your house. Get a piece of graph paper and a tape measure and get busy. After you have mapped it (maybe even put in furniture) you can go to the next level of hiding items and showing where it is on your "map". Then kids can find it. From there they can begin to map other things (classrooms, schools, grocery stories, etc.) Let them build a building (or town) and draw/ map it. Eventually, they should use physical maps and help you drive places. I know most of the world relies on GPS but you lose something if you can't "see" the way the streets are working together. For some places this is a grid - but not for my town! Helping your kids to see this can help their visual-spatial sense.
Graph Paper Games - There are MANY out there. Playing Battleship on graph paper, dots or the multiplication game (two dice and you make a box that size - whoever has the most squares after so many rolls wins - oh is that area??) are just a few. If you are going somewhere have a pencil and graph paper and play a game.
If you are truly gung ho and want to do it the "traditional way" you can get cozy with Euclid's Elements (it is public domain so you an read it for free and possibly in Latin or Greek!). Here are some great thoughts about how and where to start that at home. Knowing Euclid goes beyond geometry - it teaches you how to build any type a proof - at least that was Abraham Lincoln's experience.
Often we don't think about the role geometry plays in our lives - from mapping, to interior decorating, to proofs to fun games. There are ways to bring this subject to life and ensure your kids have a solid foundation. I'd love to hear other ideas that you have.