Hello lovely people,
There are several ways to get around Bilbao.
What I recommend most is walking. Of course like that you will be able to take in everything rangeing from impressive and contrasting architecture to secret spots, parcs and nice pintxo-bars.
What is most convenient is the metro. It goes through the whole area (you can reach all touristic spots easily) and also connects Bilbao with the ocean.
I got myself a Barik card, which you can recharge and use not only for metro but also to pay for all buslines. If you want the cheapest version you should get your hands on a personalised one. I did not (yet), because it requires some administrative work and beaurocracy but it is the cheapest option there is. I know of a friend that just took over the card of a former Erasmus student.... That's of course not legal and I think if you get caught you're banned for ever applying to one of those again but she is definately saving money.
If you can't get anywhere by metro there is a bus that fills the gap.
You reach the university campus at Leioa by bus too. (Bizkaibus, they are the green ones. Just look for UPV). There is a website that gives a nice overview over all connections by bus. Just by clicking your way through you will be able to figure it out: http://apli.bizkaia.net/apps/danok/tq/index.html?Idioma=ES