If you don't plan on using Coimbra as your stepping stone to Porto or Madrid, you could take a tour. For my money, nothing beats a guided tour on a day trip - your expert guide can give you all their local knowledge and all the logistics are planned out for you.
Alternatively, the train is a great option as it is quicker than the bus and only slightly more expensive.
Lisbon to Coimbra by Train
The train from Lisbon to Coimbra takes about 1h45.
You'll arrive in the Coimbra-B station but you can easily connect to the center of town (Coimbra-A station) via a five-minute train ride that is included in your ticket. Book from Rail Europe (book direct).
See also: Lisbon Train and Bus Stations
Lisbon to Coimbra by Bus
The bus from Lisbon to Coimbra takes about 2h30 and costs about 15€. It's about a 10 minute walk to the center of town or you can connect with a local bus. Book from Rede Expressos.
Lisbon to Coimbra by Car
The journey from Lisbon to Coimbra takes 2h and is about 200km. Compare Prices on Car Rental in Portugal
Where Should You Go After Coimbra?
Porto, home of port wine, is the most obvious place to visit after Coimbra. Read more about getting from Coimbra to Porto. Coimbra is also a good starting for getting to Madrid. Read about travel from Coimbra to Madrid.