Any selection is going to be a matter of personal taste.
If you're into history, then London, Bristol, Dudley and Edinburgh are "musts". If you are into ungulates, Marwell and Whipsnade will stand out. If you want to see bonobos, then Twycross is the only site in the UK or Republic of Ireland.
Personally, I would say that anyone from California is unlikely to be massively impressed by any UK/Irish aquarium or collections of birds or ungulates. Primates, especially lemurs and callitrichids, are another matter. On that basis Linton, Hamerton, Banham and Kessingland are probably worth a look at. These are all also advantageous in being based in East Anglia, where Titchwell, Cley, Minsmere and Hickling Broad all offer good birdingat any time of year.
The time of year is also an important consideration. The Scottish Highlands between August and March won't offer too many birds inland, other than eagles, and again a Californian would be more likely to be familiar with golden eagles than most. From April to mid June, however, Speyside would be essential. And if the time is available, an attempt to get St Kilda, one of the planet's great seabird islands and sole home to a wren taxon that might just be a full species would be a venture worth undertaking.
One final thought; the Moray Firth is home to a resident population of very big bottle-nosed dolphins, and North Norfolk will give the chance to combine good birding with grey seals, a North Atlantic species that I don't think is kept in any of the bigger US collections.