10 days in Ireland in April - where to go?

Where are you flying in and out of? It would be ideal to fly into Dublin and out of Shannon. Belfast is an easy day trip from Dublin by direct train (2h 15m).
Would you consider renting a car to explore the western side of Ireland? If so, from Dublin you can take a direct train to Galway or Cork (2h 30m) and rent a car there.

I can’t imagine getting from Killarney to either the cliffs of mohr or the dingle peninsula would be very easy with busses and trains. Dingle is wayyy closer and imo I think seeing the Dingle peninsula would be the preferred option over going very much further away to see similar landscapes in a limited amount of time. Dingle is very charming and I’m sure if you can get to Dingle you can hop on a tour of the peninsula for a decent price.

If you can I’d highly recommend budgeting to rent a car if at all possible. It’s unfortunate though because rental cars are quite expensive in Ireland imo.

I’d highly recommend Cork as what looks like, a relatively easy transit from Killarney. Cork was maybe my favourite city in Ireland as it really ticks a lot of boxes for what I’m looking for in travel

We took a train from Cork to Belfast through Dublin in 2019 and it’s quite a pleasant ride going that way. That is however a very main train line I believe

I would suggest spending your first several days in Dublin, seeing the sights there. Most of them are within walking distance, and the tram system is very good if you want to use it. You could take a day tour to a place like Newgrange (north) or Glendalough (south) if interested.

Trains to Killarney leave Heuston Station every couple of hours and take about 3.5 hours. Go there a couple of days before your conference starts and take one or two full-day tours like these. I found these by googling "Ring of Kerry bus tours," I have no experience with them but they would give you a good look at some of Ireland's best scenery. Besides the Ring, this company has a tour to Dingle and Slea Head, and another to the Gap of Dunloe in the mountains. They also have tours focused on hiking, which may or may not interest you. (Rain is a definite possibility in April.) More googling may reveal more choices, reviews, etc. This seems like the best way to see beautiful places in the southwest without driving or depending too much on public transportation. When you've done your scenic thing from Killarney, you'll be ready for the conference.

The Cliffs of Moher are great, and there are tours there from Galway (easily reached from Dublin by train or bus), but you don't really have time to deal with that spoke-and-wheel system more than once. For the same reason, I'd skip Belfast this time. It's just a couple of hours from Dublin by public transport, but deserves more than a day trip. As you must know, every time you change cities you spend a good part of a day just moving yourself from one hotel to the next, instead of seeing the sights and enjoying where you are.

Have a look at Paddywagon tours from Killarney. As well as doing the Ring of Kerry, they also do a day trip to Dingle. (They get consistently good reviews on Trip Advisor.)

Cork is about 90 minutes by bus from Killarney.

I'm afraid Cliffs of Moher is a non runner - It would take 4+ hours by bus each way...

So I did an entire three week trip by train and bus several years ago. It requires a bit of planning, but is not difficult and you will be able to see a lot of the sights by using guided day trips from the various towns. I started in Dublin then took the train to Killarney then slowly made my way back stopping in Kinsale, Kilkenny and then Galway. Since your conference is in Killarney you could start in Dublin, train to Galway, take a couple day trips from there (Cliffs, Inishmore), then train back to Kilkenny, spend a day there, then train onto Cork and take the bus to Kinsale (I did not not spend any time in Cork), then travel on to Killarney. It is easy to get a day tour from Killarney to Dingle so give yourself time for that before your conference starts. You could also do a guided day trip in Killarney Park that goes across the lake by boat the through the Gap of Dunloe on a jaunting cart. By taking short train rides from one town to the next you can cover a lot of ground, and still have time to see a lot. Just make sure on the day you travel to get the earliest train you can.