Yes. The hard part is of course to find the order of the elliptic group and its factorisation. Ideally, if the order of the group is a prime number (use a primality test), you know that k = 1, and any point on the curve is a good generator. That said, it is not because you have found an elliptic group with prime order, that it is a cryptographically safe group. For instance, if the order of the group equals p when you are working over F_p, you have a cryptographically weak curve. There are other conditions that give rise to cryptographically weak curves. Generating a cryptographically good curve is somewhat tricky. If this is for a toy example, with relatively small curves for illustrative purposes only, then as others said, use Sage.

Yes. The hard part is of course to find the order of the elliptic group and its factorisation. Ideally, if the order of the group is a prime number (use a primality test), you know that $k = 1$, and any point on the curve is a good generator.

That said, it is not because you have found an elliptic group with prime order, that it is a cryptographically safe group. For instance, if the order of the group equals p when you are working over $F_p$, you have a cryptographically weak curve.

There are other conditions that give rise to cryptographically weak curves. Generating a cryptographically good curve is somewhat tricky.

If this is for a toy example, with relatively small curves for illustrative purposes only, then as others said, use Sage.