The cheapest and easiest way to go radarspotting is using an internet connection and web browser.
There are lots of websites where you can track aircraft flying around the world in realtime. Some require a subscription but most are free and a great introduction to radarspotting. As well as tracking flights, you can check the registration and type and view a picture of the actual aircraft. Most sites will also display the flight number and the route. A pair of binoculars can't do that - yet.
And if you have a suitable phone, some websites also offer mobile apps with similar features and some offer augmented reality so you can point at the sky to check what aircraft it is and where it's going.
Moving on, for just a few Euros, you can subscribe to something called PlanePlotter and get more control over the screen layout and configure it to your personal preferences.
The PlanePlotter software runs on your local PC and you can either feed it data from your own Mode-S receiver or just view data provided by other PlanePlotter users if you have an internet connection.
If you want even more control, you'll soon be thinking about getting your own Mode-S receiver to track aircraft without the need for an internet connection, handy if you like spotting at your local airport or away on holiday (but take care as some countries don't encourage aircraft spotting!).
You can choose from a wide range of receivers including low cost kits and ready made solutions such as Kinetic's SBS or AirNav's RadarBox. Kit prices start at around 100 Euros rising to 500 Euros for an off-the-shelf solution with everything included.