For most AP mids you generally need some form of early mana regen. You can get chalice, tear or 2x / 3x dorans. All 3 options cost around 800 gold and offer good enough mana regen. However they differ in their secondary use.
Tear is for heavy lategame scaling but no early use, only get it if you plan and can play passive. Chalice is amazing mana regen and offers MR + a nice upgrade with CDR. Get chalice if you are against another AP champion or if you really love CDR. Dorans is the only choice that adds oofensive power due to the AP on it + it adds survibility vs heavy burst and is the only choice vs AD assasins for early mana regen.
The part about Dorans Ring also applies to Dorans Blade. You wanna get a second Blade if you cant afford at least a vamp on first back. If you get pushed out of lane you need more combat stats to be able to keep on fighting in the lane. Saving those 440 gold wont help you much if you get pushed out of the lane by an opponent with a second Dorans.
€: you shouldnt get 2 Dorans Shields because the passive is unique and this unique passive is basically what makes Dorans Shield strong.