In the JNCIA study guide, it is stated that there are currently 120k routes on the internet and that Juniper supported up to 450k. How did they arrive at the 120k? If I wanted to know how many exact routes, how could I find that by myself?
The backbone routers of ISPs (to be precise, Tier-1 ISPs) constitute what is known as the "default-free zone." In this zone, there are no default routes -- every route that is announced on the Internet is carried in the router's route table.
It's simply a matter of counting the table entries.
If you want to count yourself, you have to arrange a BGP peering session with your ISP and have them send you all routes.
P.S. the exact number is constantly changing as new networks come on-line or get disconnected.
In addition to the good answers already given by Jens Link and Ron:
There's no absolute, objective answer how many routes are active on the internet. The number constantly changes due to networks connecting and disconnecting, routers having outages, etc. Some prefixes are announced only to specific other networks (e.g. using a no-export community) and are meant to be seen only in specific parts of the internet.
In addition to that, there's filtering: not every network filters routes it receives the same way. Some don't at all, some filter so that small prefix (/24 upto /32 in IPv4, /48 upto /128 in IPv6) won't be accepted.
Every network has its own view of the internet. And more prefixes in a routing table isn't always better, it can also mean bad filtering, accepting a lot of deaggregated routes, etc.