If you arrive by plane:
All the information regarding how to get to/from the airport to/from the city centre is available here: https://www.aena.es/es/asturias/como-llegar/coche.html
If you arrive by train/bus:
The bus and train stations are very close to each other (the bus station is just going down the train station), and they are a 5-minute walk from the city centre. The train station, Estación del Norte, is well connected to numerous city buses (lines H-S, L-S, OVI), so if you need to move around from there, it will be very easy.
How to move around:
Whether you arrive at the train station or at the bus station, the most logical way to get to the city centre is to go on foot! The most important street in Oviedo is actually in front of the main exit of the train station, it is called Calle Uría. It is where most of the shops and emblematic (as well as beautiful!) buildings are located.
If you need to take a taxi to your hotel, there are taxi stops at the exit of both the train station and the bus station. A short-distance ride can cost around 7-8 euros.