Bathroom fan won’t turn on

The three pairs of wires correspond to the three functions. You should be able to find labels or a note in the manual explaining which is which, but for now I'll assume:

• black/white: main light
• red/white: fan
• blue/white: night light

The green wire is a ground wire.

How you wire this up depends on what you have available. A 3-way switch will not help you here; you need multiple switches, and for that to work you may need more wires between your switch box and your fan/light fixture. Most likely, you will have a single cable (with a black, white, and bare ground wire) running to your fan. In that case, you can wire it up as follows:

• bond the bare ground wire with fixture green ground wire
• bond the supply white (neutral) with all fixture whites
• bond the supply black (hot) with the fixture black, red, and blue

That assembly will work, but it's not ideal as your fan, light, and night light will all turn on/off from a single switch.

A better version would be to have two switched wires running up to the fan. You can do this either by adding another 2-conductor cable (such as 14/2WG) running up to the fixture, or you can use a 3-conductor cable (e.g. 14/3WG) where there is a single neutral (white) and two switched hots (black and red). Assuming you have the 3-wire cable, your setup would be:

• bond the bare ground wire with fixture green ground wire
• bond the supply white (neutral) with all fixture whites
• bond the supply black (hot 1) with the fixture black and blue (lights)
• bond the supply red (hot 2) with the fixture red (fan)

This allows you to control the fan and light separately with your two switches, but again has the shortcoming that the nightlight is only powered when the light switch is on. Which probably defeats the purpose!

And that leaves the final, best option: two switched hots as above, plus an unswitched hot power supply wire (let's call it black-unswitched). In this setup, you'll

• bond the bare ground wire with fixture green ground wire
• bond the supply white (neutral) with all fixture whites
• bond the black-unswitched supply with the blue (nightlight)
• bond the supply black (hot 1) with the fixture black (main light)
• bond the supply red (hot 2) with the fixture red (fan)

This setup is ideal because you have switches to control your fan and light separately, and you have constant power to supply the night light. That means your fixture can use the unswitched power to run the nightlight constantly, and when you come in and turn on the main light, the nightlight either turns off or just becomes irrelevant.

As you can see, you have a few reasonable options for how to wire this fixture. Another variation is that you might have unswitched power in the fixture area, and only one cable running to your switch box. If that is the case, you'll be using the cable to the switch to both send power to the switch and receive power back when the switch is on. If you have that sort of set up, you could provide constant power to the nightlight even if you have a single switch controlling the fan and light together. You'll need to make a couple more junctions in/near the fixture in this case, involving the switch.

Note: a 3-way switch will not be helpful. A 3-way switch has three wires (call them A,B,C) and allows you to switch between connecting A+B or A+C. Two 3-way switches can be used together to allow switching a light from two locations. But as long as you have a switch with only two positions, you can only choose between on and off, not off, on-fan, on-light, and on-fan+light. For that you need two switches, or some sort of switch designed to control two separate fixtures.

Final cautionary note: To do this right, you will need some basic electrical knowledge, including how to properly connect wires with wire nuts and how to work on a fixture safely. If you aren't familiar with this sort of electrical work, you should get someone who is (experienced DIYer or pro) to work with you. Improper electrical work has a massive risk of hurting you, either while you're trying to install it or later on when a failure can put you at risk of fire. Good luck and be safe!

Without actually being there, and not being able to see the wiring at each of the devices, it's difficult to be sure what's going on. But here's my best guess, based on past experience.

To make it easier, I've numbered the ungrounded (hot) conductors in the photo from left to right.

1. You've already established goes to a light.
2. Supplies power to the switches.
3. Likely goes to the fan.
4. Also goes to the fan, but since the fan doesn't have a light built in, it's likely capped off at the fan.
5. Probably feeds some other device, maybe a receptacle.

Why 3 and 5 are controlled by the same switch, I have no idea. Again, I can't see the room, so I have no idea what equipment you have.

If 1 and 4 are both for lights, it makes sense that a singe switch controls them.

One other thing to check, is to make sure the terminals of those switches are rated to accept more than one conductor. There should be a label on the switch that tells you what type, and how many conductors can connect.