After spending two months in Cork, Ireland, I feel like I can provide good advice for a nice day tour in Cork. I stayed in July-August, so some of my comments may be affected (by good weather, mostly). Here is the photo album of the whole time.
I think the best way to go around is on foot, as you can really enjoy the town, but there are the Coca Cola bikes that can be rent for 3 days: you just need to register online, pay a few euros, and then pick a bike in the many bike points scattered around Cork, and return it in any other one.
A day tour in Cork
If you want to go to the tourist office, for example to take a map or ask for some information, the best time to do so in when you get in town. In fact, the office is near both the bus station and the CityLink stops. In the building just in front of the tourist office, at the second floor, there are free toilets, handy in any circumstance!
In Shandon (St. Anne’s Church) you can ring the church bells! It really is amazing! Music sheets are provided with popular songs (Hey Jude, Lord of the rings soundtrack, and many more) and the bells sequence to be played. It is a bit out of the centre, but it does even go through nice streets. For students, the entrance is just 4 euros, and you can also get to the top of the bell tower and see Cork from above. Beware that the last entry is early in the afternoon, at around 16. This church is also called The Four Liars because, with strong winds, the four clocks on its sides are said to never display the same time.
Exactly on the other bank of the river there is Elizabeth Fort. It is not much, they are basically some high walking paths, but again you can see Cork from above (although not as high as the Church), and it is free! Closes at around 17. Near here there is also St. Fin Barre’s Church, which is patron saint of the city. The church surroundings are not bad, the church is huge (but entry is not free) and there is a very disappointing labyrinth.
On a clear day, Fitzgerald Park is a beautiful place. It’s the biggest park of the city, there are often events in the weekend (music/festivals). There is even a bar that makes nice launches for reasonable prices (~5 euros for a sandwich), and you can then eat on the grass on the river banks. It’s a bit out of town, like 20 minutes on foot, but you can go there basically all through pedestrian-only streets, and there is a walking route that goes along the river bank on the other side: it starts from Shandon’s side and is called something like Banks of Lee Walkway.