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(1) In general, an ideal is infinite, so it would not make sense to give a complete list.
Do you have a particular setting in mind where ideals are finite?
(2) To add ideals, just use the plus sign.
Example.
sage: R.<x, y, z> = QQ[]
sage: G = R.ideal(x^2)
sage: H = R.ideal([y^2, z*z])
sage: J = G + H
sage: J
Ideal (x^2, y^2, z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field
