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.
The four liars, St. Anne's Church, Shandon
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.
Fitzgerald Park, Cork
Even if you don't need any groceries, the English Market is a great place to visit. Besides all the stalls that sell sweets & chocolate, there are all local farmers and it's very nice to go around. It is located right in the centre and has several paths in and out.
While going around the centre, my favorite narrow street can be accessed by North Main Street: it's a tiny hole going towards the river: you will go through a narrow path all painted by local guys. A spot of love.
A beautiful hidden narrow street in Cork
For a snack, there is an awesome place called Sticky Fingers that serves a stunning variety of donuts for 2/3 euros, and they are really good! It's just in the centre, opposite to the Opera House, besides a Costa Coffee.
If you happen to be in town later than 18, the "Sin è" pub always has some great live music and a lively atmosphere, peculiar of Irish pubs, and is definitely worth a visit. If it is crowded, you can get into for the music and don't even take anything to drink. This pub really offers the true Irish Pub experience that I have not felt in many other more mainstream places (like The Oliver Plunkett). What I have found to be the best burger in town is offered by West Cork Burgers, 2 minutes from the centre. A nice place to hang out and have a pizza is Tom Barry's - the pizza is actually quite good, and is not expensive - lots of young people there. Finally, a good italian option is Ristorante Rossini.
Also, always go around and look for street art: it's everywhere! Rubbish bins, houses, electrical boxes, walls... anything that can be decorated is likely to actually be so!
Street art on house in Cork
A nice bookshop, both for second hand and new books, is called Vibes and Scribes, just 5 minutes out of the centre.
If you happen to be in town at sunset, and it's a good day, on the river walk that heads away from the centre starting from Fitzgerald Park you may be lucky enough to appreciate one of the most beautiful sunsets. There, I witnessed one of the best sunsets I have even seen. The walk will take you at least a full hour to complete.
Sunset on Lee River Walk, Cork
Trips from Cork
If you are in town for more time, the trips I can recommend are:
Galway, the Aran Islands and Cliffs of Moher. This is probably the best I had, plan for 4-5 days. Inishmore is the biggest of the Aran Islands and is amazing. You can get a bike for around 10€ for the full day and go around it. My strategy has been: every time I came at a junction, to where fewer people went. I ended up in the middle of nowhere, truly breathing the island atmosphere. Be sure to bring your lunch and anything you need for the whole day. Go to Dun Aengus as well: if you can't make it to Cliffs of Moher, you will see some real irish cliffs here!
Inishmore, Aran Islands
The Cliffs of Moher is another beautiful place, but packed with tourists. I took a hostel in Lisdoonvarna, a desert place, then took a bus to Doolin early in the morning and walked from there to Cliffs of Moher, slowly rising and climbing up to the top of them. Don't get fooled in the day trips that take you to the tourist center of the cliffs and back - have the full experience.
Cliffs of Moher
Connemara is another nice region around Galway, but I didn't have time to explore it much. And Galway is nice too, although quite touristic. The Aran Sweater Market is a great place to buy real Irish wool, very warm indeed!
The Beara Way. A ring that goes all around the Beara peninsula. It crosses some very nice villages, all quite still living the real irish life, and full of so many kind people. Be warned that getting good weather in that place is really a jackpot: most of my days there were spent walking in the mud looking at my feet, cursing at all the goats that seemed to at ease in that inhospitable land. Beautiful sceneries when the sun was out, though.
Allihies, Beara Way
Killarney and its National Park. This was a bit disappointing for me. I expected a big national park, and I found the part I went into packed with tourists. The city of Killarney was full of tourists and of shops for them. I stayed one night and went to Ross Castle, Muckross Abbey and Torc Waterfalls, but they were all full of people and nothing really exciting.
I have had the feeling that the nice part of the park started after Torc Waterfalls, which is the farthest I could go without missing my bus back home. Overall, I did not find it much worth for a day visit, but I met a canadian guy who had walked the Ring of Kerry, which goes deep into the park, and was amazingly happy!
Killarney National Park
Kinsale. A nice small sailors village on the see to the south, just an hour bus from Cork. There are some nice shops, while the beaches and nice nature is a bit away from the city (around 30 minutes walking at least). May be worth going to Hold Head Castle, but is pretty far from the town itself and really requires a means of transport (around 20 km, mostly on concrete).