Working out automorphism groups of algebraic varieties I came across the notion of the automorphism group of an algebraic variety. I was unable to calculate those groups for very basic examples, such as the sphere and the circle.
Is there any intuitive way to deal with and work out such examples?

I'll assume we're working over $\mathbb{C}$ for simplicity - I'll also assume that the "circle" you are talking about are the varieties
$$ X = V(x^2 + y^2 - z^2) \subset \mathbb{P}^2$$
Recall that the rational map $\mathbb{P}^1 \to X : [u,v] \mapsto [u^2 - v^2, 2uv, u^2 + v^2]$ is in fact an isomorphism.
Thus $\operatorname{Aut}(X)$ is isomorphic to $\operatorname{Aut}(\mathbb{P}^1)$.
Fact: The automorphism group of $\mathbb{P}^1$ is $PGL(1, \mathbb{C}) = GL_2(\mathbb{C})/\mathbb{C}^*$ where a $2\times 2$ matrix acts in the obvious way (N.B. these are the Mobius transformations).
You can probably find a proof of this in multiple places. Hartshorne I Ex 6.6 will walk you through a proof.
In general (Hartshorne II Example 7.1.1) one can show that $\operatorname{Aut}(\mathbb{P}^n_k) = PGL(n, k)$.
