Annoying dots on pictures

As others have said it's most certainly dust or dirt on the sensor. What hasn't been mentioned is the reason your second picture is clear of dust is most likely because you used a larger aperture. The smaller the aperture you use the more dust shows up, and with reasonably wide apertures it is simply invisible or barely visible.

The easiest way to check for dust without special equipment is to use a long lens (like your Tamron at 300mm) to eliminate any distracting elements from the frame, point your camera at a patch of clear blue sky or a uniform white surface or similar, use the smallest aperture possible (such as F/22 - F/32) and take a couple of shots. (I generally take two and point the camera at a slightly different position in each to make it easier to distinguish between dirt on the sensor and any irregularities in what I'm photographing).

The K-5 has a menu function that will map out the dust for you, not sure if the K-x does, but in any case it usually doesn't show everything up and I find the above method better.

If it's just a speck or two which barely show up I usually don't worry about it... after all how often in reality do you take photographs at such a small aperture? If there are a few more and/or they're a bit more visible then I will give the sensor a clean.

I also got dust into the viewfinder the first time I used a Rocket Blower and had to have it professionally cleaned to remove this. However since then I have learned to use it correctly and have not had any further problems whatsoever. It's a shame they don't come with any instructions on how to use properly but the method I was shown (by a photographer far more experienced than myself) and have used ever since with no problems at all goes like this:

• Hold the camera with the open lens mount pointing directly downwards.
• Hold the blower underneath the camera, pointing directly upwards.
• Do not insert any part of the nozzle actually inside the camera body. Keep it fully outside the body leaving at least a centimetre or two clearance.
• Squeeze the bulb firmly to produce short but even, controlled bursts of air. Do not squeeze it so hard that all of the air is expelled in an instantaneous violent blast.
• Allow the nozzle to move slightly from side to side whilst squeezing the bulb so the flow of air can reach all parts of the sensor, but try not to let it point anywhere else in the camera body other than at the sensor. (This might sound difficult considering you can't see the sensor while doing this, but really it is just a matter of familiarising yourself with the position and size of the sensor beforehand and using a little common sense).

Personally I use the blower first when cleaning the sensor. If this doesn't sort it out I use the Arctic Butterfly two or three times. If there is still dust remaining after that I wet clean using the orange V-swabs and VDust Plus fluid. As Smeggy says you could just use the swabs every time but since they are fairly expensive and not reusable, trying the dry methods first can save you a bit of cash in the longer term (not so much in the short term because an Arctic Butterfly brush is not exactly cheap).

Of course, you should always try to minimise the amount of time the open camera body is exposed when changing lenses to help minimise the accumulation of dust in the first place. Personally I use the white plastic body cap which came with the camera to cover the body whilst putting the first lens back in my bag and getting the next one out, but everybody has their own methods when it comes to lens changing.