Coimbra is 100 km away from Porto and 200 km from Lisbon, both cities having an international airport. Although Lisbon has more direct connections around the World, Porto also offers direct connections to many international destinations.
From either airport you can use:
Train
The metro (running every 5-10 min in Lisbon and every ~20 min in Porto) to the railway stations of Lisboa-Oriente (Lisbon; 5-10 min) or Porto-Campanhã (Porto; ~ 40 min), respectively.
There you take the national trains ("Alfa Pendular" (AP) or "Inter-cidades" (IC) to Coimbra (1h35 to 1h50 from Lisbon; or 1h from Porto), that depart every hour on average.
Shuttle
Service door to door by: www.airportshuttle.pt
Bus
Rede Expresso, if departing from Lisbon airport: select Lisbon (Oriente). You can reach "Oriente" by metro (3 stops from the airport station); if departing from Porto, select Porto (Aeroporto).
Flixbus, if departing from Lisbon airport: select Aeroporto de Lisboa, if departing from Porto, select Aeroporto Francisco Sá Carneiro.
Car
Not necessary in Coimbra but if you choose this option expect about 1h ride from Porto and 2h from Lisbon to central Coimbra.
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Each city has its own public transport system. If you need to use bus in Coimbra each ride is 2 euros (buy ticket from the bus driver) or you buy a yellow SMTUC card (1 euro) and charge it with 10 rides (0.64 euro per ride).