I just got my first 4* (Fireshrike). Does the 4* rarity mean I should focus on her rather than any red 3*, or are some 3* better than 4*? Will I need a lot of training and a duplicate Fireshrike to make her more powerful than all 3*?
3* dragons are needed in some trust events and basic bout quests. You will need a team of 3* dragons. But first of all you need a strong team. If you can be sure to get enough (another one to train it to 3* and one more to train it to max) 4* dragons to train those to max, you should first focus on 4* dragons, otherwise level up 3* dragons first. And although fireshrike is kind of fine, not all 4* dragons worth leveling up. You can read the description of their abilities and see if it can fit in your team.
To fully level a 4*- or even 5*-star it will cost you fish, scales, dragons & doubles. Gold later as well.
Fish is the most missing at the beginning and later comes also money - by now I reached the point scales and money is the most missing to level up my stronger dragons.
Like Wodensfang wrote: Some specials require 3*dragons so these ones still come in nice to have when levelled up even if you have stronger ones in your team.