I recently started getting into Go, and think I now want to build

Well, whatever you do entirely depends on what equipment you have, the kind of effort you want to put in it, and the amount of money you are willing to spend on it, be it materials or tools.

If you are feeling really cheeky and want to recycle some board scrap of the same thickness, you can always use dowels to join them together, plane them flat and finish. While it won't look absolutely amazing, some can argue that it can add "character" to the piece, but I don't mind. If you want something nicer, look for woods that please you - cedar, oak and even cherry. Wood choice is up to you, and that will affect cost and finishing methods. Some woods, depending on where you are, can be hard to source for whatever size you are going for because not many may be willing to cut a large piece for you. Feel free to prove me wrong, as I am unaware of the availability of many woods worldwide.

As for gridlines, depending on the wood, you can route them out, use a sharpie or even burn them on. All require different equipment, have varying results and need varying levels of experience. Burning a pattern on always looked nice to me. Whatever you do for this step will determine the kind of finish that is available to you.

As for finishes, you can always oil it up with linseed, apply polyurethane, waxes, leave the wood alone, and my personal favorite, shellac for a french polish. Whatever you have for the gridlines may have a bad effect on the finish, such as bad visibility, so take that into consideration.

I'm a big fan of french polished goods, especially when it doesn't fit the item, so I'm a bit biased towards that. I suggest you look into shop-cut woods or even just wood availability in your area and start your search there. MDF with a layer of stencilled in paint and clearcoated might just be the cheapest option I can think of, apart from using a pencil for lines.