Firstly, you gotta remember SNAAP - That is, Sodium, Nitrate, Ammonium, Acetate and Potassium are always soluble (That is, any salt made with these are always completely ionised with exceptions below).
Then, anions that are always soluble include Chlorine, Bromine, Iodine (The Halogens) and Sulphate. The exceptions to these are that Silver and Lead cations for ALL of these are either insoluble or slightly soluble (You would say in an exam though, they are always insoluble). Then, Sulphate has additional exceptions of Calcium and Barium.
Anions that are insoluble include hydroxide, sulphur, carbonate and phosphate - All of these have the exceptions of Sodium, Potassium and Ammonium EXCEPT FOR HYDROXIDE which is soluble with Sodium, Potassium and Barium (Slightly soluble with calcium)
This is frequently all you need to know, but also be very aware of Barium Sulphate (Colour, common reactions). They love to ask about it in exams.