What you were taught in your catechism classes is indeed still valid, although it is not commonplace today to hear the description of the various states of the Church as you describe them.
The church triumphant describes the Church in heaven. In the kingdom of God — the realm in which the holy Trinity, the angels and saints, and the abode of all those who have reached the fullness of salvation in Christ — the Church already exists. But it will have its fullest being at the end of time, when all of creation and (we hope) all human beings will be conformed to Christ and all reality will be one of divine praise and glory.
The term “church triumphant” underlines the truth that in the glory of heaven all human sin will have been transformed, death and suffering will be no more, and the glory of God will have triumphed over all the imperfections of human history.
The church militant refers to the Church on earth. The term “militant” can suggest an antagonistic relationship between the Church and the world. Nevertheless, it refers to an authentic reality: that the Church on earth works to overcome the imperfect and sinful dimensions of human existence.
The Church’s mission is not to oppose the world and society, but to work for their transformation by the convincing preaching of the Gospel and by the edifying power of the good works and example of those who are baptized into Christ. The Church’s best asset is the saintly activity and example of those who have chosen the Christian way of life.
The church suffering refers to the church in purgatory. Purgatory is not a kind of temporary hell. It is rather the threshold, the antechamber of heaven. In purgatory, all those — whether Christian or not — who have reached the gates of death without reaching the full perfection of life represented in Christ are cleansed in a kind of ongoing baptism and are purified by the enlightening fire of the Holy Spirit. The suffering of purgatory is not one of destruction, but the suffering that comes from leaving the old self behind and taking on the new.