That is a very ambitions itinerary if you are relying on public transport...It is doable but not easy.
You don’t say how you are getting from Ireland to Scotland.
For convenience, I am starting from Glasgow.
To reach Mull you need to catch the train to Oban and then catch the ferry to Mull. There is a bus that takes you from Craignure ferry terminal to Fionphort where you catch another ferry to Iona.
Calmac timteables here.
There is no direct ferry from Oban or Mull to Isle of Harris and to be honest this is a nightmare journey. The easiest way is to get the ferry back to Oban and then train back to Glasgow. The 916 bus goes from Glasgow to Uig on Skye which is the ferry terminal for Isle of Harris.
Leaving Mull, catch the ferry back to Oban. You then have to decide how to get to Isle of Harris. The easiest is to catch a train back to Glasgow and then catch 916 bus which takes you to Uig on Isle of Skye which is the ferry terminal for Harris. This will involve an overnight in Glasgow and very early start in the morning. It is 6.5 hours on the bus too.
The other alternatives involve catching a bus to Fort William and then train to Mallaig, ferry to Armadale, public transport across Skye to Uig for the ferry.
OR catch a train from Glasgow to Inverness, then train to Kyle of Lochalsh and then bus to Portree and onto Uig ...
Bus services on Skye aren’t brilliant and more geared to getting people to and from work.
Once you get to Harris the bus service is better.
Rather you than me. I would seriously rethink this plan....