If you want to explore something a bit different than international chess, I’d highly recommend taking a look at Xiangqi (Chinese Chess). It’s a wide-open and more unpredictable game, focusing more on tactics than strategy.

You’ll notice some interesting features, starting with the board itself. The board is considered to be 9×10, since the pieces are placed on the line intersections. In the middle there’s a river that affects gameplay, and each side has a 3×3 palace out of which their Generals (king) cannot exit.
The Red side I’ve always seen colored as red, but the Black side can be black, green, or blue. In fact, green’s been used in about a quarter or third of the sets I’ve seen.
The pieces are represented by a Chinese character, and sometimes, different words are used for different sides! This set is unusual in that most of the time, the Cannons (located on the rank behind the pawns) usually use slightly different characters. My set does have a picture representation on the back, but it’s not helpful in the long-run because most sets don’t have anything except the Chinese!
The Cannon is a Rook-like piece that moves like a Rook, but to capture, it’s required to jump over exactly one other piece (friend or foe).
One of the most striking and important aspects is the lack of a solid rank of pawns. There are only five, which means there’s a lot of wide-open action pre-made for the Rooks and Cannons. “Pawn structure” is meaningless, and according to various sources, there’s not even a real struggle for control of the center. (If you recall, in chess, the Rook is the only piece that can control as many squares from the corner and side as from the center.)
There’s even a “fool’s mate” in Chinese Chess that’s four moves long (I lost to it my very first game), where both Cannons are lined up on the same file as the general.
Some links:
Peter Donnelly gives an excellent introduction to Chinese Chess here.
Normally, I don’t like linking Wikipedia, but their Xiangqi article is very well done.
A lot of basic strategy and tactics here. The English is imperfect, but comprehensible.
This guy has a freeware Xiangqi program Qianhong (According to the site: The name Qianhong (”chyen hOng”) means “Light Red”. It is a play on the name of IBM’s “Deep Blue” Chess computer and refers to this program’s weak computer AI.)