They might do it for free... they might charge you $50. Cleaning it yourself is actually NOT particularly difficult.
I use a four-phase approach:
Remove the lens and...
1) point the camera at the ground, activate the camera's self-cleaning cycle. Re-check for dust. You'd be amazed how many times a simple dust-cleaning cycle WHILE the camera is pointed at the ground will just shake it off.
2) using the menu, activate the manual sensor cleaning mode. This simply causes the mirror to swing clear and the shutter slides open. What you see behind is REALLY the filters which are in front of the sensor (it's not actually possible to touch the sensor itself... there are IR/UV filters in the way.) Use an air blower -- this NEEDS to be the type which is a simple bulb you squeeze to blow puffs of air. Don't use cans of compressed air as these can do two bad things... first, they can spray fluid onto your filters -- which just make a mess, and second they have a chilling/freezing effect as the gas decompresses. This chilling will attract condensation to the sensor... which also just makes a mess. But a manual blower that you squeeze will work great. Point the camera at the ground and give it a few puffs.
Frankly... after doing #1 or #2, my sensor is always clean and I never have to proceed any further. And note that I haven't actually "touched" the sensor (really the filters in front of the sensor) with anything. This is totally safe.
3) If it still has spots, use a VERY soft brush. You can buy a paint-brush for this or a use a "Lens Pen" sensor cleaning brush. Resist the urge to feel how soft the brush is by feeling it with your fingers... that'll just get your skin oils on the brush. Give the brush a couple of sharp whacks on the edge of a counter or desk to knock off any dust that might be on the brush, then very gently brush the sensor to remove the dust (by the way... there are versions of these brushes that are grounded to release any "static cling" in case particles want to stick to your sensor due to static.) This is VERY gentle and safe.
Odds are, you won't have to proceed any further, but there's one more trick just in case the previous three steps failed.
4) If the dust is still there, you may need to use a magnifying loupe to inspect the sensor and determine what is so stubborn that it won't come off with the previous 3 methods. I have a scope for this (and they're cheap). I'd suggest you buy some "Sensor Swabs" and "Eclipse" sensor cleaning solution. Both are made by a "Photographic Solutions" and they're probably the most popular cleaning. The Eclipse solution is methanol with a very high purity level -- so high that it doesn't leave any residue on the sensor. Follow the directions. You'll put a few (2 or 3) drops of the solution on the swap (never put the drops straight into the camera), then gently wipe the sensor and whatever is there will come clean. This stuff works EXTREMELY well and is designed specifically for this purpose. The reason I suggest you use a magnifying loupe or scope to inspect the sensor is to make sure there isn't something that you don't want to drag across the sensor (e.g. dragging a piece of sand across the sensor under pressure could cause a scratch -- remember it isn't *really* the sensor, it's really a UV/IR filter and if you did scratch it, it could be replaced... but that's a hassle, so better to inspect and be safe.)
I have NEVER needed to go beyond step #2. But then I am conscious to protect the opening to the camera body every time I remove a lens to avoid letting anything get into the camera body.
I recall (and it's been a while) _some_ D7000 owners complaining that Nikon had been just a bit too liberal with the lubricate on the mirror hinge and this could splatter onto the sensor. A standard cleaning cycle, or puffs of air, or even a brush would not remove oil splatter. For that you'd need to use the Eclipse fluid -- and you'd probably NOT want to use the brush as that would just get oil on the brush and smear the oil around the sensor.