The first step in combating sensor dust is understanding it. The dust is much more visible at smaller apertures than it is at larger apertures. At first, it seems odd that your aperture would effect the appearance of dust on the sensor. If you think about it, a small aperture works like a small light source and a large aperture works like a large light source. With a small aperture the shadow created by the dust is very hard, making the dust stand out. With a large aperture the shadow created by the dust is much softer and so the dust spot isn't as visible.
To see how dusty/dirty your sensor is, the best technique is to take a shot of something light and smooth/featureless (a completely blue sky, a white sheet of paper) with the aperture stopped down as small as it will go. If you can't find anything perfectly featureless, shoot something close and put it out of focus. The goal is to get a picture in which the only features are the dust spots.
Now that you can see what dust spots you have, you need to know how to get rid of them. Before you start with that, a few words of caution are in order. The biggest danger I've seen in cleaning sensors isn't scratching the sensor - it's wrecking the shutter. The shutter has to be kept open to give you access to the shutter. If it closes while you are working on the sensor, it's very likely that it will be damaged. I don't know about Nikon's, but with Canon's the shutter closes when the camera looses power or gets shut off. For that reason, I always recommend cleaning the shutter on a fully charged battery. Some people recommend using the AC/DC adapter that comes with the camera, but I'd rather not gamble against a power outage.
There are four different ways of cleaning a sensor - blowing of dust, brushing off dust, and wiping it off. Newer cameras often have built in sensor shakers that also help, but you either have that and don't need to worry about it or you don't have it.
Your first cleaning effort should be to just blow off the sensor. If you can clean it without touching it, you virtually eliminate the change of scratching it and reduce the chance of making it dirtier. Go to someplace clean and dust free. Put your camera in sensor cleaning mode, hold it upside down (so more dust doesn't fall onto the sensor), and try to blow the dust off. Use a squeeze blower or a mini-fan. DON'T USE COMPRESSED AIR CANS as they often have oils and propellants that can make things worse. The idea is to gently blow off any dust or heavier particles.
If blowing didn't clean your sensor, the next thing to try is brushing. I recommend a brush designed for sensor cleaning rather than just grabbing a paintbrush at Hobby Lobby. The problem with paintbrushes are that they often have sizing to stiffen the bristles and this can leave a residue on your sensor. It helps if you charge the brush before using it because it will attract dust better. You can do that by wiping it quickly on something clean and dust free (this is like rubbing a balloon on your hair to build a static charge) or you can blow air through it with your blower. Once your brush is charged, make a couple of wiping passes in the same direction and then check your camera again.
If you have dust spots that are still there and haven't moved, you've got to wipe them off. If they have moved, I'd just try brushing again. To wipe one off, you want a cleaning pad on a relatively stiff handle and some sensor cleaning fluid.
You won't find, at least for Canon, any officially endorsed products for sensor cleaning. Their position is that you should have it cleaned by a certified tech. That may be safer, but it's way to much hassle for me. I've been cleaning my own sensors for 5 years and have never had a problem.
I have used on other product that is very convenient for sensor cleaning. It's called the Dust-Aid. It's basically a slightly sticky foam pad on the end of a little plastic wand. You stick it against the sensor and then pull it away. The idea is that the dust will stick to it and get carried away. It works very well, but it only replaces the brushing step. It won't help if you get goop on your sensor. I'm also concerned that frequent use might eventually make your sensor sticky.
It's hard to get a sensor totally and perfectly clean. My goal is clean enough. When shooting portraits or sports (usually with wider apertures), I just shoot for not having any big spots. When shooting macro and landscapes (usually with smaller apertures), I'm quite a bit pickier.
I think the site with the most comprehensive guides and products for sensor cleaning is
Copper Hill Images.