There are thousands of types of multimeter. I don't know which one you have, or what the readings mean without seeing it.
It sounds like it is the fuse, but to make sure, could you post pictures of the meter in the following conditions:-
1) low Ohms, no connection
2) low Ohms, meter leads connected together
3) low Ohms, meter leads connected left and right prongs of power lead with the switch on, selector set to 110 V
4) low Ohms, meter leads connected left and right prongs of power lead with the switch on, selector set to 240 V
5) high Ohms, no connection
6) high Ohms, meter leads connected together
7) high Ohms, meter leads connected left and right prongs of power lead with the switch on, selector set to 110 V
8) high Ohms, meter leads connected left and right prongs of power lead with the switch on, selector set to 240 V
(please make sure you say which photo is which)
and make sure you don't touch the meter leads, especially in the high Ohms range, as many meters will measure your body resistance.