I was told that this is a sequel to the Silent Hill games so obviously I'm going to play SH 1, 2, 3, and 4, but what other games do I need to play to be up-to-date?

None really. Every game besides 3 and maybe Origins is it's own thing and doesn't need the others to know what's going on.
You'll be fine playing Downpour with no prior knowledge of the rest of the series.

Downpour has absolutely nothing to do plot-wise with any of the other games in the series. It it completely stand-alone and can be played by a total virgin to the franchise.

Ok that's great, I'll get on it when I finish the main SH series (I've already started playing them and am now on the 2nd one; they're EPIC!!!).

Yeah, what the others said.

Downpour was actually the first game I had my friend sit down and play. She got through it without asking too many questions/getting lost, and it even got her interested in the rest of the series.

A lot of the easter eggs and extra references to past games may not mean much to you if you haven't played them, but it shouldn't hinder your understanding of the overall game.

True, Downpour is stand-alone, but if you wanted to know all of the games before Downpour:
SH1
SH2
SH3
SH4
SH: Origins
SH: Homecoming
SH: Shattered Memories

And I'd recommend playing all of them at some point if possible. In fact I always recommend playing all games in any series in order, simply to get the original experience, but that's just a preference. In my opinion, each SH game is definitely worth playing though.

I probably wouldn't recommend Downpour as an introduction to the series for someone because they might end up thinking the series spends a ton of time on random sidequests, but it's not your first, so no problem there.

The previous games follow a semi-concrete timeline that can be ascertained, for the most part, by tracking the rise and fall of the Order. SH0, SH1, and SH3 follow a direct plot, while SH2 fits in there somewhere. SH4 mentions plot points from SH2 and SH3, which means it had to have come after them, and Homecoming also mentions some things that occurred in SH3. I'm not sure where it fits in, but Downpour most likely occurs after Homecoming since all the previous games seem to have occurred more or less in chronological and numerical order.

So I think it's safe to assume Downpour is the most recent, chronologically speaking, in the timeline.

But yes, it can definitely be played as a stand alone title.