A UV filter won't actually do anything to improve your images because your camera already has a UV filter built into it (there are actually two filters directly in front of the sensor -- one of which deals with the UV light.) So a 2nd UV filter on the end of the lens would just be redundant. That said... there are those who just want it for protection.
Cheap filters can do some detrimental things to your images. I've seen reviews of UV lenses that tested how much UV they block as well as how much ordinary visible light they block. A few are appallingly bad... blocking very little in the UV range, but meanwhile blocking a bit of visible light that you didn't want blocked. They can create unwanted reflections resulting in flare, ghosting, and reduce the optical quality of the image.
Before you drive yourself crazy scrutinizing image quality reduction caused by filters... if these are just web-size images and you're taking them for fun, then go ahead and get a cheap filter. Yes... it will degrade your image quality. But at small web-size image resolutions, it's REALLY hard to notice the difference. On the other hand, if you're trying to shoot a high quality image, then consider using high quality filters or just remove the filter completely and let your lens go naked.
Circular polarizers are one area where I don't necessarily encourage being cheap. A bad circular polarizer can do a lousy job blocking reflections while at the same time doing wonky things to your colors.
If you don't want to buy filters for each unique lens diameter you own, then buy the filters for the largest diameter you need, and then buy "step up rings" to adapt the filters to the smaller diameter lenses.