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That $E$ has self-intersection $-1$ is Hartshorne Proposition V.3.1. The relation between effectivity and ampleness is more clear in the case of divisors on a curve where a (non linearly trivial) divisor is effective has degree greater than $0$ if and only if it is ample (Hartshorne IV.3.3). A So certainly for curves effectivity implies ampleness. On the other hand, even on curves, there exist ample divisors which aren't effective, for example consider a smooth non-hyperelliptic curve of genus at least $3$ and take the divisor $2p-q$ for $p,q$ points on the curve. This is ample but not effective.
As for how to think about ampleness in general, a divisor is ample if and only if some tensor power of it is very ample (Hartshorne II.7.6), and very ampleness is convenient to think about since it basically says you have an embedding into projective space and that the locally free sheaf of rank $1$ associated to the divisor is the pullback of $\mathcal{O}(1)$ from the projective space. Finally, a curve might have negative self-intersection only with itself, so you can still rely on Bezout's theorem working as your intuition does! Also, as you can see, all the above statements are in Hartshorne!
That $E$ has self-intersection $-1$ is Hartshorne Proposition V.3.1. The relation between effectivity and ampleness is more clear in the case of curves divisors on a curve where a (non linearly trivial) divisor is effective if and only if it is ample (Hartshorne IV.3.3). A divisor is ample if and only if some tensor power of it is very ample (Hartshorne II.7.6), and very ampleness is convenient to think about since it basically says you have an embedding into projective space and that the locally free sheaf of rank $1$ associated to the divisor is the pullback of $\mathcal{O}(1)$ from the projective space. Finally, a curve might have negative self-intersection only with itself, so you can still rely on Bezout's theorem working as your intuition does! Also, as you can see, all the above statements are in Hartshorne!
That $E$ has self-intersection $-1$ is Hartshorne Proposition V.3.1. The relation between effectivity and ampleness is more clear in the case of curves where a (non linearly trivial) divisor is effective if and only if it is ample (Hartshorne IV.3.3). A divisor is ample if and only if some tensor power of it is very ample (Hartshorne II.7.6), and very ampleness is convenient to think about since it basically says you have an embedding into projective space and that the locally free sheaf of rank $1$ associated to the divisor is the pullback of $\mathcal{O}(1)$ from the projective space. Finally, a curve might have negative self-intersection only with itself, so you can still rely on Bezout's theorem working as your intuition does! Also, as you can see, all the above statements are in Hartshorne!