the sigma takes 77mm filters.
Most modern lenses - the Sigma included - already has UV filtering in the glass. Getting the UV filter is really getting more of a "scratch prevention filter" as cheap as possible
The Sigma - even at it's widest, doesn't grab the filter edges. When lenses do, it looks like vignetting, but it is not - although it looks the same. If the filter is more than 7.5mm thick there can be some such "vignetting" in the 10-11mm range of the lens. With the 19mm thick filters, there can be some in the 10-13mm range.
As for the types of filters to get, there's always the standard polarizing filters and the neutral density grad filters to deal with very bright skies.
There are of course a gazillion other filters of all colors, stars, softening whites, etc. etc. but that is part of your creative process and only subject to your tastes.
Nikon 12 - 24 mm f/4
Tokina 12 - 24 mm f/4
Tamron 11 - 18 mm f/4.5 - 5.6.
To consider, if you haven't already. However, the Tamron and Tokina would only focus manually on your D40x.
The Sigma can have some non-linear distortion that can be difficult to correct in post-processing. Horizontals can show "humps" as opposed to a much preferable smooth curve.
You may also have to prepare yourself to send it back once or twice, if it's not completely sharp when you receive it. The quality assurance and build-quality on this Sigma explains why it's cheaper than the Nikon. On the other hand, if you can get it sharp (or if it is when you get it) it's a great deal.
If you're like me, you treat your gear like a Fabergé Egg - pros bang it around with impunity and expect it to last.
I would like to get my hands on this lens as well for my D40 and I look very much forward to seeing some of your shots when you have had it out for a spin.