Save for your faves. If you don't have any favorites yet, save anyway because you might end up liking someone that you see while you play. You could also just go for one of the best servants in the game (Merlin, Waver/El Melloi, and Scathatch=Skadi) whenever they're available (though I think that the third one is story locked).
Spartacus, Georgios, and Hans are all welfare servants worth leveling, and they all serve different purposes (farming/wave clearing, stall/defense, and buffs/crits/healing respectively).
If you want to save for specific classes, sabers are going to be really rare early on, since there are only 3 available to you at the start and all 3 are 3 stars. Rulers and Foreigners can also be useful, but those are kind of rare.
Oh, and don't forget to put things in your support team.