Wiring bathroom exhaust fan / light
The two-way switch in the picture switches a single conductor on and off. I expect you will find that the switched black wire and a neutral white wire lead up to the ceiling fixture. With this setup it's not possible to separately control the ceiling fan and light. But it should be possible to turn them both on and off together. It sounds like that's what you're trying to do.

The exhaust fan should have come with some instructions on how to wire it. We expect to find a neutral white wire, two hot wires (black and blue?) and a grounding wire (green or bare). If the existing wiring to the ceiling is non-metallic cable then you connect black to black/blue, white to white, and green to green/bare or grounding screw. If the existing wires are run as metallic shielded cable then the shield provides grounding and you will need to bond the grounding wire of the fixture to the metal ceiling box.

It sounds like you've attempted the first scenario but the fan is not switching on? Some bathroom exhaust fans are setup to run for a fixed period of time after detecting a certain level of humidity. These can be finicky and may not always turn on when switched on. You might read over the instructions to see if this is what's getting in the way. If it's a simple fan + light then tying the hot wires together should cause them both to turn on. Check your connections and if you can't get it to work try connecting one at a time to debug. It's possible the fan is a dud out of the box.