As mentioned in the comments, the additional content immediately appears at the places on the map where it can be accessed. However, it is useful to know where to find them if already have explored a lot and don't want to backtrack.
Grimm Troupe
Apart from the minor improvements (e.g. charms), the major part of the Grimm Troupe update is available after 'summoning the troupe' at a small cave accessible from the large cliff on the west of the Howling Cliffs. After that that, the Grimm Troupe appears in Dirtmouth.
Hidden Dreams
The Hidden Dreams content doesn't really have a quest like the Grimm Troupe, but a noticeable piece of content is the Dreamgate (unlocked after gathering enough Essence) and some two bosses in dreams that are hidden:
1:
"Grey prince Zote". Found in Bretta's house. Unlocked after both saving Bretta and defeating Zote in the colosseum. This of course requires you to not let Zote die when you encounter him, so this is the only part on which you could miss out in your current play-through.
2:
"White defender". Found in a hidden room near the Dung Defender. Unlocked after defeating the Dung Defender and all three Dreamers.
I struggled with this particular attack for so long until I stumbled upon an awesome way to dodge it by accident. Since this technique helped me so much, I wanted to share it in case it could help someone else.
Basically, the monarch wings are reset every time you boop someone on the head, so you can take advantage of this to stay in air above him for a really long time. It also slows the pace of the fight a little bit, giving you room to breath.
Try doing this:
When he first comes towards you, jump up as high as you can (potentially with your double jump too).
When you fall down, he will probably be right underneath you. Use this time to downward slash at Zote. When you hit him, it will reset your monarch wings, so you can jump up again. This allows you to stay up in the air for a really long time.
Since this is somewhat hard to describe, I made a video of me demonstrating this technique (ignore the fact that I died :P):
You can see me doing this technique at 0:50 and again at 1:17.
One thing to keep in mind when you're trying this is that sometimes, Zote will trip just before you would have slashed at him, so you miss him on the floor and then fall into him. Staying up as high as possible will help minimize this possibility, but you may need to use your dash at the last second to avoid falling into him.
To maximize the effectiveness of this approach, I highly recommend Longnail/Mark of Pride to make downslashing easier, and Dream Shield so you can get a little bit more damage in while doing this.
Best Answer
Currently (version 1.0.1.4) there are three endings and one post-credits scene that can be unlocked. The endings are in order of obtainability (named after their associated achievements):
Finally, the post-credits scene is obtainable on any ending by talking to Mr. Mushroom seven times in his different locations with the Spore Charm equipped. It shows Mr. Mushroom flying over Dirtmouth while staring at the camera ominously before ending on a "To Be Continued" screen. The associated achievement for this ending is Passing of the age.