All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
First of all, we congratulate you and your mother on embracing Islam, which is a great blessing that requires showing gratitude to the Provider, Allaah The Almighty, by remembering and thanking Him, and by learning the Islamic teachings, constantly observing them, and accompanying the righteous Muslim women.
As for your mother, you should try to convince her to travel with you to live in that Muslim country, explaining to her the great benefits it would achieve to her religion. Seek the help of Allaah The Almighty in convincing her. If she is convinced, then all perfect praise be to Allaah; otherwise, you may hire a maid, preferably a Muslim and trustworthy woman, to serve her.
As for traveling with your children to a Muslim country, it involves obvious religious benefits, which include being near your husband, guarding your religion and your children's, and distancing yourself and your children from the causes of temptation. Indeed, guarding one's religion is given precedence over any other interest.