What else can be done to level up besides fetch quests as a low level resto-druid?

Try to do balance. If you are a restoration it is hard to deal out damage. If you want to level I recommend doing dungeons and going to goldshire to ask for help doing them. If you are 13 you can dungeon so I would recommend grinding levels until 13 and then dungeon

I'll try and answer each of your questions one at a time:

When will i be able to play a dungeon?

You can join a dungeon in 3 levels, Beginning at level 15 you can open the dungeon finder (default key is Ctrl+I i believe), which will send you into a dungeon with other random people playing the game to complete dungeon quests and level up fighting bosses.

When will I be able to play with my friends?

You should be able to play with them now, depending on what they want to do!
If they are all doing the end game content, you're going to have to level up to the same level as them, and buy the newest expansion.
However if they don't mind exploring and leveling with you, they can make a new character and get to where you are and you can adventure together! I've done this with my buddies and it is good fun.

Is there anything besides fetch quests that I can do at a low level?

There should be a slight difference in quests you complete, such as take this item to this location, collect a certain number of skulls or tusks or feathers from this animal, etc. If you let us know what race you started as I could see what island you are on, and there are plenty of leveling guides online that could show new quests you maybe have not yet found.

What is the difference between buying BfA and only paying the monthly fee?

If you start paying the monthly fee, you'll be able to level up all the way to level 110, and have access to all the content up until then, including all the expasions with the last one being Legion. Upgrading out of the starter edition includes quite a few other things, a list can be found here.
Buying BFA just means you can level up to level 120, and have access to all the Battle for Azeroth content, which is where all the end game content now resides, and where most of the players are.
Depending on how fast you level, you may want to wait on BFA, as there is a new expansion coming in 2020, and once that launches the $15 a month should, but is not guaranteed to, include BFA since it will no longer be the "newest" expansion.

As for leveling up while in Restoration specialization, that can be challenging.
I would recommend switching to your Guardian specialization, you can change at any inn, and leveling up with that. The Guardian is a tank spec, so you will take reduced damage and not die as easy, as well as being able to output adequate damage, you should have a spell or two to heal you as well (I have not played a druid in a looooong time so I'm a little foggy)