Steveo covered most of the West Cork bases, knows more about the scenic spots than I do tbh. I'd echo the get a car comments if you're heading southwest out of the city. Bus services between the the towns and villages in West Cork are sporadic to non existent outside of the big towns (sadly the West Cork Railway died a death decades ago), car is really your only option if you want to visit the rural beauty spots.
The city is great, small enough that you'll do it all in a weekend.The usual thing of museums, good food, reasonable night life. Cork City Gaol probably the highlight museum wise. There's Elizabeth fort as well which is in the city limits.
Then you have the big tourist draws like the Blarney stone which sounds a bit shit tbh and its outside the city,The English Market if you're a foody and the usual mix of pubs and restaurants for all tastes. One of the big tourist draw is apparently the Butter Museum, rather random, never been myself.
There's a Titanic/immigration museum down in Cobh which is cool, Fota Wildlife park/nature reserve thingy as well which you can wander around (Cobh train goes to both). Random scenic walking routes in the city include out down the harbour to the Blackrock observatory (which has astronomy stuff if you're into that kind of thing and also a bar). You can go west from the city centre out to UCC (Glucksmann Art Gallery), the Mardyke, Fitzgerald park (cool small museum there) and onto the Lee Fields. If you feel adventerous you can trek up to the northside by Sundays Well where you'll get great views of the City and harbour.
If you're basing yourself in the city then the likes of Kinsale, Crosshaven and Clonakilty are all close enough to the city to visit from, are pretty/scenic and have stuff of interest, and have decent bus connections to/from Cork.
If you're a history buff then let me recommend two sites in particular; Spike island is a former Navy base and prison, its situated in Cork Harbour and has a cool imperial history going back centuries, great views as well from it, Likewise Charles Fort in Kinsale which has some great history and views (Kinsale town has Desmond castle as well). You can get to Spike via ferry from Cobh, Kinsale harbour is beautiful and the town itself is a great food and night spot.
Loads of natural beauty like the ones Steveo referenced, most well away from the city though - Garnish island, Bantry Bay, Mizen Head, the Beara Peninsula, Allihies etc. Stick in and around the N71 road if you are driving and that'll take you near the scenic spots.
Most importantly, If you want a football fix then the League of Ireland will be in full flow in April. Local behemoth Cork City FC play on Friday nights and the atmosphere is class, especially when Dub scum teams come to town.