Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$.
It would be very instructive to see some explicit examples, but I don't know where to begin.
For instance, let $E$ be given by $y^2 = x^3+1$ and let $\ell = 5$. How do I see directly that $H^1(E,\mathbb Q_5)$ is semi-simple?
Have you tried looking at Serre's "Abelian $\ell$-adic representations and elliptic curves"? That might shed some light.
If you just want some individual examples, you can look at elliptic curves with CM [which includes the case $y^2 = x^3 + 1$ you gave above, as Lubin's comment points out]. Here you can literally write down the Galois representation; it's induced from a character $\psi$ of the Galois group of an imaginary quadratic field and the irreducibility is immediate from the fact that $\bar\psi \ne \psi$.