There's also a "Popular" tab (the one with the star), where you can see the most liked photos on the network right now in a big grid of thumbnails. (Note: This tab is usually pretty dumb.)
The third way is the most complicated, but it's also the most interesting. The "News" tab (the one with the heart in the chat bubble) shows you all the notifications about liking and following. Within that tab, there are two more tabs. The "Following" side shows all the photos liked by people you're following.
This is the best way to discover new photos and the people who took them. If you find a photo in here that you like, you can tap the name to see the photographer's profile. If you like what they do, follow them, and now you'll see all their photos.
The other side of the "News" tab is the part that's actually called "News." That's where you'll see all the likes and comments on your own photos. It also shows when new people follow you.
It's always fun to check out the photos of the people who follow you. You can follow back the ones you like from here.
The fourth way to discover photos in Instagram is using hashtags. Hashtags included in the caption of a photo are clickable, so you can see all the photos that have that tag. You can also search for hashtags. So if you want to see pictures of cats, you can search for '#cats.'
The Point
In short, the point of Instagram is to make connections with people who see the world in interesting ways. You can find and follow people based on the kinds of images that inspire them, and if they like yours, they'll follow you back. The simple mechanisms of liking and commenting provide great fun and feedback. It's a new kind of network that's perfect for the smartphone age. No wonder Facebook bought it.
You can download Instagram for iPhone or Android for free at Instagr.am.