What should I save for?

I don't know if anyone can give you an honest answer to this question.
What to save for comes down to personal preference and servants you genuinely want to roll for. As a F2P player myself, i am patiently saving my quarts for what i want to roll for, i don't throw it around on every banner in an attempt to "fish out" a rare servant i don't particularly care for much. It's completely fine to skip banners you don't like.

You got a lot from the Tutorial Summon, considering that at least one 4 star is guaranteed. I myself got Marie and Siegfried.

I don't know what's in store for NA in terms of upcoming banners but i bet that you can go to https://www.reddit.com/r/grandorder/comments/7nrzy5/jp_banners_since_launch_v2/ and look up what banners might appeal to you. We're in the second picture, Waver and Iskandar banner is the current one. The next two, Cu Alter and Medb were previous America banners so we're skipping those and going straight to Kintoki and the rest.

Like the guy above me mentioned, that's a tough question to answer since the servants you want to pull for really does come down to who you wanted to begin with.

Myself, I'm a big Fate Extra fan so I originally started playing this game to obtain the different variations of Tamamo and Nero and then proceed to build teams around them. It's not really important for me to actually get the strongest guys in this game.

If you're not really a fan of anyone specific in the game and just want strong servants in general, you could use this as a general reference. Note however that this was just something I fished out of Google, but I generally do agree with what was listed in it, even though I won't really be rolling for a lot of those servants myself because of the reasons I outlined earlier.

https://www.reddit.com/r/grandorder/comments/7noqfl/top_10_servants_to_save_for_in_2018_fategrand/

Also note that some of the servants in there have had banners already, but they'll be reappearing again in banners later on. Jeanne Alter in particular will come back later this year, so if you really want those hard hitters, you'll have another chance at her still.

Save 300 saint quartz~

There are 2 ways on hoarding and spending SQs in this game: for waifus/husbandos and for good servants.

Personally, I'll test my luck by summoning 1 10x summon on the Raikou banner coming soon then just save for Merlin.

Remember that having an all-powerful arsenal of godlike servants personally isn't that important, as you have the friend support system to help you get through the game. Some servants from the FP summons are also incredibly good, such as Medea, Robin Hood and Hans, to name a few.