First time sharpening, do I sharpen on the coarse or fine side? I have a Japanese Shun chef's knfve, and I've been bad about sharpening it. I haven't sharpened it since buying it in August and now it takes a bit of effort to slice through a potato or carrot. 

I just bought a Japanese Water stone on recommendation from my butcher, it's dual sided with 1,000 grit on one side and 4,000 on the other. The advice I was given, was to sharpen it on the fine side (4,000) weekly and the coarse (1,000) ever few months. Given, that I've never sharpened it since buying it months ago, should I start on the 1,000 side and then give it a go on the 4,000 side ?
Yes. The coarser grit removes material faster, but leaves a rougher finish. The finer side then allows you to bring the edge to a fine polish. The most important thing to remember when sharpening is to keep the blade at a constant angle relative to the stone -- don't change the tilt of the blade as you move or you'll just make a mess of it.