How do I install a bathroom exhaust system?
Congratulations on your new place! Installing a bath fan is a somewhat ambitious project for a novice DIYer. You have to:


Do electrical work to wire the fan and switch.
Mount the fan in the right location. 
Run ductwork from the fan and outside, usually via a roof or wall vent.
Avoid doing much damage to your ceiling when cutting into it.


The biggest challenges here are usually in the vent installation. If you can go out through a wall or gable, a vent can be easy to install, just requiring a round hole and some caulk. If you have to go up through the roof, there are a number of subtle ways you could install the vent incorrectly that would not be immediately obvious but would increase your chances of a future roof leak significantly. Try searching the site for "roof vent flashing". Also, depending on the height and angle of your roof, you should think about whether you're able to climb up safely.

The other potential issue is the electrical. Like roofing, it's easy to make a mistake that you might not notice for a while, but the risk is more severe as you could create a fire. You could also make a mistake that shocks and injures you. So if you are really new I'd recommend you start by learning how to safely do things like replace outlets and light fixtures first.

Most of all, make sure you take the time to understand what you are doing. A challenge is always fun, but you will probably have plenty more opportunities!