The official Warframe site[warframe.com] contains the latest news.
The official Forums[forums.warframe.com] are a great place to talk about it.
Twitter will keep you up to date as well.
And let's not forget about Reddit!
For some spicy memes see /r/memeframe

These resources allow you to keep up to date with various on-going events in game. Such as the day/night cycle, invasions, daily sortie, and more.
https://hub.warframestat.us/ is a very nice site that shows almost anything you'd want to know.
Baro Ki'Teer/Void Trader[warframe.fandom.com] wiki page. It shows a timer and, if he's live, what he has in stock.

Furthermore check out the Warframe app on iOS and Android to see alerts, start and claim items from your foundry, chat with your clan, and more.

The Warframe Wikia[warframe.wikia.com] pretty much contains everything you need to know.
Warframe Builder[warframe-builder.com] is a nice site to, well, make builds! You can view the Forma required and all the stats to make your desired build.
https://overframe.gg/ is a more modern website to create, share, and search for builds.
Story and History[warframe.wikia.com]. A long read but interesting nonetheless. If you care about lore and such. Be wary of spoilers. Read only the intro and Orokin Era parts if you want to avoid any (major) ones.
List of all Warframes[warframe.wikia.com]
Cetus/Fortuna map[semlar.com] along with fishing locations and info.
Semlar also offers a comprehensive overview of the Riven price history[semlar.com] from the trading chat.

Warfame Market[warframe.market] is an online (3rd party) auction house for all warframe items.
Or use the XB1/PS4[forums.warframe.com] and PC[forums.warframe.com] trading sub forums.
There's also a subreddit: /r/wartrade

There are a ton of Warframe YouTubers who showcase the latest gear and new builds. But let's start of with the PlayWarfame channel. Here DE uploads their Devstreams, Prime Times, trailers and Tenno Reinforcement videos.

Tactical Potato shows all the latest and greatest, along with builds and thoughts. Nothing too advanced.
StallordD makes interesting lore videos and some SFM animations. He doesn't upload an awful lot of content but most of it is rather informative.
I know there are a ton of other channels. Great ones and not so great ones. I just listed some I know, watch and like.

There are a ton of Warframe streamers as well but I hardly ever watch Twitch streams. There is, of course, the official warframe channel[www.twitch.tv] for Devstreams and Prime Times.
For other Twitch streamers I suggest checking out the Fan Resources[warframe.com]. Here you'll find more sites and YouTube channels as well.

Did I forget anything? Is there an amazing site in existence I missed? Any general questions? Please let me know.

28/06/2020 Wow, I actually updated some parts of this after however many years.