How do we wire a new bathroom exhaust fan to our bathroom?
There's several parts here
Can I tap into the junction box for power?
Yes, provided it's not overloaded with wires.
How do I wire it?
If I were you, I'd add a second junction box. You'll have

2-line power from the existing box
2-line cable run to your fan
3-line cable run to the switch

Find the breaker to the box you want to tap into and turn it off. If it's 15A, buy 14 gauge wire (Romex is white), if it's 20A, buy 12 gauge (Romex is yellow). Be sure to clamp the cables if any box is metal.
Inside the new box, wire

All 3 white wires together
All the bare grounding wires together (run an extra one to your box grounding screw if the box is metal)
The incoming black wire from the other junction box to your 3-line black wire
The black wire going to your fan to the red wire

Inside the junction box for your switch, it sounds like you have an existing switch. If so

Break the tab on the side of your double-switch
Wire the old switch wire to one of the switches
Wire the new black and red to the other switch
Cap your new white wire with a wire nut