Path of Fire (PoF) gives you access to mounts. You will need that at a minimum if you want them. They are extremely helpful, even in the old world.
Heart of Thorns (HoT) has a ton of content and meta events that people like to farm. I definitely recommend it. Most people go back to HoT zone metas after finishing PoF. And both expansions have elite abilities that are very powerful, if you raid, PvP or WvW you will want them.
There is also extra content called Living World (new story and zones). It comes in episodes that you can pick up free for a bit when they are first released. After that you have to buy them.
Living World Season 3 (1 can't be done anymore, not sure about Season 2) is the content between PoF and HoT. It is a continuation of the story and has a bunch of new zones. The story leads up to HoT's. You can buy the season on the aution house.
Living World Season 4 is finishing up right now. You can get the last episode for free if you log in sometime in the next week or so. You have to buy the earlier episodes.
The Raptor and Griffon are the most used mounts, they (and 3 others) come from PoF. But there are 2 mounts, the Roller Beetle and Skyscale, that come from Living World Season 4. You need to own and do those episodes to get them. The Roller Beetle is from episode 3 or 4, you'll have to buy it to get that mount. Skyscale comes from this latest episode that is free atm. There is also a WvW mount, but it's useless in PvE. Looks cool though.
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As for what to do, GW2 has so much content that unless you can play it 24/7 it's pretty hard to do all of it. I pick a goal (get a specific mount, build a legendary, get more mastery points, level a new toon, etc) and then just work towards that. With so much going on in the world and people all over, it's very easy (and fun) to get distracted. So my goals tend to take a while to finish.