http://www1.toronto.ca/wps/portal/toronto/portal?vgnextoid=6bf6d1e80065c210VgnVCM10000067d60f89RCRD is the city's tourism site; there are plenty of others with advice on activities and attractions. By the start of June it should be warm enough for restaurants and pubs to open their terraces. There could be outdoor concerts but indoors the spring season of plays and museum shows will be wrapping up. Some of the live theatre action will have shifted to the Stratford Festival but Soulpepper, in an old distillery district, keeps repetory plays going through the summer. www.soulpepper.ca At any time of the year Toronto is a great city for movies, now augemented by screenings at the permanent home of the Toronto International Film Festival. Check www.tiff.net for attractions; the four daily newspapers in English carry extensive listings.
If you really want to be idle in the outdoors, old-fashioned ferries waddle across the inner harbour to the parks and beach of Toronto Islands. http://en.wikipedia.org/wiki/Toronto_Islands On a crowded summer day you can see the truth of Peter Ustinov's old statement that multicultural Toronto resembles New York, only run by the Swiss (although it is not so neat and clean as when he said it....)