Get a blower, and be sure to exercise reasonable precautions when changing lenses, although "reasonable" need not be taken to such a degree that it really hampers your shooting. In general get out of windy conditions and, if you can, angle the camera down when changing the lens. Of course, if you can get in a car or indoors, it's nice, as long as it doesn't take you away from your photography.
Then, if you've followed the advice to get the Gitzmo Rocket Blower or another good blower, you give it a few good blasts in sensor cleaning mode without making contact with either the mirror or the sensor, and then you'll likely face two facts:
First, the blower really can dislodge loose dust! As much as you can hold the camera face-down while doing this so that hopefully it will trickle down.
And then, you'll need to go farther than just a eyeball visual inspection.
There are two common ways of checking for small particles on a sensor: First there are "sensor loupes" that can both light and give a magnified view of your sensor. I've never had one but hear they are quite effective.
And then Second is the "acid test". For this you need a scene that you can fully frame (at whatever focal length you need) that is a solid light color. My favorite for this is a clear blue sky, but a light overcast could work or, maybe a white sheet.
Set your lens to MF and your camera to a narrow aperture and your WB can be, well, something that doesn't really mess up your color, although if you shoot Raw it doesn't much matter, and in the daytime Auto WB handles things quite well. With the 5D setting your aperture to f/16 is nifty. Since your lens is in MF you can fire a shot with nothing in focus, and, in fact, you don't want anything in focus. Any detail at all, even the weaving on a white sheet, can defeat you, so just adjust your MF to keep things OOF -- for this we only care about what's on the sensor, and it is the narrow aperture which will best show up dust and such.
Take the shot, or as many shots as you want, hand-held is fine!
Then, bring the shot(s) into your image processor.
Now, you will view them at whatever magnifications you wish, but at some point you'll want a reasonably magnified view. Whether or not you go to a 100% view doesn't matter so much, you just don't want the image so shrunken down that any spots just disappear. So, as big as you'd like but not necessarily so huge as to make "spots" just a blurry mess. And, make sure you look at every portion of your frame!
What you will likely find, since your sensor was obviously not clean before, is that there are a number of spots on your screen. Maybe a lot, maybe not a lot. Maybe big and ugly, or maybe small and light enough to make them invisible at a wider aperture and a normal view. Likely it will be a combination of the above...
But, if not now then eventually you will be faced with the challenge to tackle "real" sensor cleaning. After using a blower, eventually you will find particles that aren't loose enough to blow off.
At that point, there are several common solutions -- a couple involving either a "sensor brush" or a "sensor pen", and then finally the "wet cleaning" approach that to some is scary but is actually the one thing that "really works", so be prepared.
So, let me give you three company names that all are involved in this stuff. Check out their sites, read their info, look at their products, and be prepared to go with "stuff".
They are:
Photographic Solutions, makers of the Eclipse sensor fluid and their make of sensor swabs
Copperhill, which has extensive writups and available "stuff"
Visible Dust, which provides kits for both dry-cleaning and wet-cleaning (and a blower) and includes a little gadget called the Arctic Butterfly which spins a sensor brush over the sensor which can dislodge some "stuff".
But, since you are asking about the 5D Classic, I should give you a heads-up:
Back when the 5DC was in its youth, people bought it and eventually had to tackle the sensor cleaning thing. Back then, the Photographic Solutions outfit with the Eclipse fluid and sensor swabs had become quite popular and "standard", because their stuff worked.
But then, some bad news: a significant number of 5D users found that when wiping their sensor with Eclipse on a swab, the filter over the sensor would get all messed up! The very high alcohol content in Eclipse had, in other words, started to dissolve the filter!
That caused obvious consternation with both the company and with Canon and they began consulting. Canon embarked on a fix in the factory. Photographic Solutions, on their part did two things: first of all, they promised to replace any sensor "wrecked" by the Eclipse solution, and second they put out a new version of Eclipse (Eclipse2) which was guaranteed to not mess up that filter.
Now, all that happened in the early days before I got my 5D, and by then the word from Photographic Solutions was that Canon had "fixed" whatever in their process was responsible for putting out faulty sensor/filters. And, over time the issue went away, and the original version of Eclipse was declared "safe" for use of any current/newer body.
What I don't know is what range of serial numbers might be susceptible to this problem. When I bought my 5D in '07 everything had settled down. I bought the original Eclipse with my 30D but then used it with my 5D Classic and have never seen a problem with it.
But, people occasionally pop up who typically are buying a used older 5D, and ask about sensor cleaning, and even though I still recommend the Eclipse method it is with this warning story.
So, I'd check out your body as to the date and the possible problem. Either Contact Canon Service or Photographic Solutions with your serial number, and they should give you some proper guidance!
