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I was looking to get a circular polarizing filter and an UV filter for my Canon 550D, 18-135mm lens. When I checked online I found these filters came in different sizes, like 62mm, 67mm, 72mm, 77mm. How do I know which is right?
Also, will there be any degradation of photo quality if I put both these filters on the lens?
As per other answers, it is usually marked with the ⌀ symbol on the front and, if not, on the barrel. Some specialty lenses do not accept filters, in which case you won't find any markings.
For your lens, the thread is 67mm
This is the thread size which means you can attach that size of filter directly. This convenient but costly. Instead, I buy my filters in the largest size (77mm usually) and have step-up rings to bridge the gap. A step-up ring costs about $12, so if you buy 77mm filters and have a 58mm, 67mm and 77mm lens, you need 2 step-up rings: 58->77 and 67->77. The only catch that you can't use a step-up ring and a lens hood at the same time. It saves lots of money considering a good polarizer costs over $200. Even if you have just two lenses with different filters you'll save. My lenses have 8 filter sizes so you can imaging how much money I saved on polarizers alone!
There will be a degradation in image quality if you use a filter. See my answer to this question. Generally, the less you pay, the more degradation there will be. UV filters are usually sold for protection but polarizers have a genuinely useful photographic purpose, attenuating glare, surface reflections and increasing color saturation of the sky and some other surfaces as a side-effect.
Take a look at the business end of the lens, i.e the end you point at subjects. Around the outside of the glass is a plastic bit where it says 'Canon Zoom Lens 18-135mm' etc.
Somewhere on there will be a symbol - a circle with a line through it, followed by a number. In the case of your 18-135mm, it should say 67mm.
The filter size is usually printed on the lens. If you look at the front of it, it's usually either just inside the edge before the glass (which I think is the case for your lens) or just along the outside of it. I suspect it's 67mm, but you should confirm it. It will have what looks like an O with a vertical line through it (Greek phi), followed by the size in mm. Something like Φ67mm is what you want to find.
Matt linked to an answer about UV filters in terms of effect. In general, the more glass you put in front of the image sensor, the more you impact the image quality. However, this can often be undetectable and acceptable for the gains that a particular filter may provide in other ways. For example, a polarizer may allow you to get a shot that you might not otherwise, so if there is a loss of image quality, you'd accept it. Having said all that, it's not really an issue for the most part. The only thing to be aware of is that polarizer will reduce the light by a couple of stops.
Your question about diameter has already been answered I see. As for the stacking question, stacking filters should be avoided unless absolutely necessary. Each filter adds two new surfaces that can be dirty and/or scratched, which is a bad thing. Each filter also adds two new surfaces which can cause flare and reflections, which is a worse thing.
As a general principle, ditch the UV filter. It serves absolutely no photographic purpose on a digital camera. It gives some small measure of protection for the front element, but so does a lens-hood which in addition does serve a photographic purpose by reducing flare. Add a filter only when you need to - the only ones you need to, largely speaking, are polarizing filters and neutral gray filters.
No matter what filter size you use, you always want to put the polarizing filter on last. The end of the polarizing filter rotates so that you can filter the light from any direction (or camera orientation), so you don't want to have anything on the end of that.
I'm not even sure if my polarizing filter (B+W) has threads to accept additional filters stacked on top now that I think about it.
Check the inside of the lens cap, it should have the filter diameter.
If you put both the filters on the lens, you may get black corners on your images at 18 mm.
