yes, you can attend Mass in any Catholic parish, even an Eastern Rite if any are around you. You are automatically a member of the parish in whose geographic territory (boundaries) you reside. You really should give your financial support to the parish you are going to attend, where your children will be educated and receive sacraments etc. If you wish to do so you should register in the new parish so that your contributions will be recorded for tax purposes. One parish will maintain your sacramental records, and that is your baptismal parish. If your child is baptised at St. Mary’s, but later receives first communion and Confirmation at St. Michael, they will notify St. Marys so that a note is made in the baptismal register.
Please seriously consider attending mystagogy sessions in the original parish for at least a year. Experience has shown that when new Catholics do not participate in these sessions and do not become part of parish life, they are much more likely to abandon the faith.