После suggest используется герундий, после offer инфинитив.
1) Mary suggested us studying English together
2) Maysa offered to buy a dictionary
3) Jeher suggested us going to the park by the metro
4) My friend offered to take another piece of cake
5) My friend offered to let them rest for a while.
6) Mother offered me to have another cup of tea
7) one of the pupils suggested us visiting Jennet
8) Monitor of the Group offered to arrange some ex. during the winter holidays