What do I buy after Dragons of Stormwreck Isle

I have just finished the Dragons of Stormwreck Isle set with my sisters, and wanted to play a new adventure, but I didn't know what to use next. What sets or books do I buy? The Dragons of Stormwreck Isle was my first time as DM, as my last experience was playing through part of the Essentials kit with my stepdad and older sister, me and my sister being the players.

What sort of adventure does everyone want to play? Princes of the Apocalypse was one of the first 5e adventures and is easy for a newbie dungeon master to run. Dungeon of the Mad Mage is another dungeon crawling adventure that will take players from level 6 to 20. 

If you want to create your own adventurers then you just need the Player's Handbook, Monster Manual and Dungeon Master's Guide if you want more than just the basic ruleset. 

Phandalin and Below: Shattered Oblisk is a good choice. The first half is the much loved Lost Mines of Phandelver campaign from the original starter kit that's ideal for newer DMs and players and takes you from levels 1-5 and then it expands with a sort of second campaign from levels 5-12.  Otherwise as Rathkryn says a lot of it comes down to what sort of campaign you want to run

I finished the starter set with my party as the DM, is there an extension to this? or was the starter set just an intro into the actual game of DND, i don't know if I'm supposed to continue the starter set...

Stormwreck Island was pretty much what you say, a short intro to D&D. There’s no extension to it but as it only goes up to level 3 almost any campaign starting at level 1 can be adapted to follow it up just by skipping the first couple of level up milestones. A lot of prewritten campaigns also start with a short level 1-3 intro that can just be dropped to allow you to tack it onto the end of what you’ve done 

oh okay, thankyou so much... am I as a DM allowed to add onto it myself though by extending some features, for instance, it makes note of Shurrath possibly appearing some day again... could I make my own sort of extension including topics like these?

You're the DM, you can do whatever you like! 

Seriously though if there's a logical follow up you think you can work out then go for it. It'll be worth watching some YouTube videos for advice on structuring a campaign because a plot isn't all you need (I like a channel called Mystic Arts for those sort of videos but Matt Colville is also a huge help) but you'd also be surprised how little you actually need for a campaign. The hardest part is pacing it and filling up individual sessions with stuff for players to do

Dragon of Icespire Peak, AKA the Essentials Kit which came before Stormwreck Island, had three expansions that took it from 6th up to 12th level.