If you do go to Iona, don’t worry too much about making transport connections. The buses on Mull wait on the Oban ferries, and vice-versa when going from Mull to Iona.
Although, to be fair, it’s only the second best rail journey in Scotland.
Yes, the journey from Waverley to Haymarket is quite impressive!
That’s third!
Ooh, I’m finally qualified to answer something on the dope! Although I’m originally from Edinburgh, so my advice would be to get on the first train to the east coast, and start your sightseeing there 
I believe the museum you’re talking about is the Peoples’ Palace. Not actually been there, but maybe I should…
I used to live in the West End of Glasgow, near Byres Road, so if you’re looking for good restaurants, you’ll be falling over them there. I particularly liked The Bothy, in Ruthven Lane. It serves modern Scottish cuisine.
Will your cousin be providing help getting around while you are there? Could they drive you places? It might make it a little easier for you to see more of the place. As mentioned, the Burrell Collection is a good bet, and it has a good cafe, so you could make a morning or afternoon of it. It’s in Pollock Park, so if it’s a nice day, you can have a lovely walk too.
I was asking my husband (a native), and he suggested having a look on Hidden Glasgow for something that interests you. I know that you can get a glimpse of abandoned train platforms from within the Botanics (near the entrance at the top of Byres Road), which is quite spooky. I’m sure there are more little gems like that to be found, if that kind of thing floats your boat.
If you have any more specific questions about Glasgow (or Edinburgh), I can maybe answer them. Without knowing what interests you, it’s quite hard to come up with things that I’d be happy recommending.
Oh, and May might be a really good time of year to come and visit. For the last few years, May has been one of the best months. But then again, it’s Scotland, so who knows what this year will bring.
In the most remote parts, it’s true that postbus services exist, and can be a great way to get an introduction to local people, but for god’s sake don’t start trying to get random posties in Glasgow to give you a lift, not if you value your life!
http://www.royalmail.com/portal/rm/postbus gives the relevant info (I did, just to be sure, check for ones in East Anglia…)
In Glasgow there’s a steel-hulled sailing ship, the Glenlee, built in 1896 moored on the Clyde which is open to the public.
The only comment I have to make is that May is possibly too early. There’s an excellent chance of lots of bad weather in Scotland. You may want to wait until June or September.