Welcome to the forum. 
GD is designed around doing the main campaign 3 times - Normal, Elite and Ultimate - though it’s possible to get to L100 on Normal if you want to.
The Shattered Realm you can do at any point once you’ve killed Kreig at the end of act 1 base game; the Emissary appears at all the major hubs and you just talk to him and then follow him through his riftgate. Of course diving into SR early will be difficult, but you can try it out and see what it’s like if you want to.
You can farm monster totems for loot and when you become hostile with hostile factions like the Beasts then their Nemesis bosses will appear. There are also what we call Celestial bosses that you can run into via quests or accidently if you’re not careful.
What is also possible is to get a character into either Elite or Ultimate Forgotten Gods content and there you can buy Merit Tokens for your other characters to use. These unlock higher difficulties so you can try starting on them if you want to.
If you have the Crucible battle arena add on you can start an L1 character in main campaign, exit and hop into the Crucible to get a Merit Token and use it, then exit back to main campaign and choose the higher difficulty to start from L1. It’s not easy, but it can be done if you want to.