> I've been told that the 18-135 lens isn't very good
Actually that isn't the usual reputation of this lens at all. It may not be built for the ages, but it's actually quite good optically.
> have seen that in some of my shots (especially when almost fully zoomed out)
Before assuming that it's the lens, it might help to investigate why the shots didn't come out as well as you'd like. A sample with EXIF information would help. To put it bluntly, most of the time when someone has bad results with a lens generally regarded as pretty good, it's the user's technique that's at fault.
> is the 55-200vr the optimal second lens for sports and shots where I need the zoom?
The 55-200vr is a nice lens, but it depends on what kind of sports you might be considering. I can think of quite a few sports situations where the 55-200 will fall far short of what's necessary. Any sports indoors will likely not work at all with this lens, and any sports at night is almost certainly out of the question.
It also depends on whether or not 200mm is enough focal length for the application. (Zoom refers to the variable length nature of the lens, not the actual amount of magnification.) Say a bit more about what you'd be shooting. There are some things for which 200mm is plenty, others where you'll really prefer something longer.
> Another option is the sigma 18-200mm lens, but unsure if the quality of that is on par with the nikon.
In some respects it's better. It's definitely better built than any of the Nikkors you've mentioned. (It still falls well short of the professional Nikkors in this category, though.) Optically it is close to a match for the Nikon 18-200VR, and certainly for most applications it's functionally as good.
_____ Brian... a bicoastal Nikonian and Team Member
My gallery is online. Comments and critique welcomed any time!