I made a new character on Skyrim, and I'm trying not to use any potions at all, using restoration healing spells instead. What perks do you recommend I take to maximize healing output, and what does restoration do when you level it up? Please be detailed.
The most immediate effect of levelling Restoration is that it gives you a discount on the magicka cost to cast a spell. Further, you have to level it to meet the minimum skill level for the more advanced perks (some of which give you even more of a discount on magicka casting cost), but I'm guessing you already know that.
Unfortunately, without more information there's no way I can really give you a definitive build strategy on what perks to get. "All of them!" is what I would say, but that's only if you have the perk points to spare. How you play, and what other perks in the other skill trees you want/need for your build has to be considered in order to find the optimal set.
Personally, I would go for the magicka-reducing and healing-boosting perks first, since that's your focus: Novice Restoration, Apprentice…, Adept…, etc.; Regeneration; and Recovery 1 and 2. Ward Absorb I'd avoid if I don't plan on using Wards and take soon if I do. Respite is useful if you're going to melee, and of limited use if you're not. Avoid Death can be great, but only if you don't regularly get one-shotted since it only triggers if you go below 10% health without actually going to 0; it would be a third-tier pick for me, not first or second. Necromage is similarly situational: if you use the undead-fighting spells a lot, then great, but if you're really only going for Healing spells then skip it. Finally, Restoration Dual Casting is great if you like dual casting spells and not great if you don't. This seems to be a matter of taste. It's a third-tier pick for me, as in I eventually get it just because I don't have any other perk I want more.