For this kind of question, resrobot is an excellent tool.
It appears the quickest transport is a combination of train and bus:
- Train Stockholm – Norrköping; for example, 07:59 - 09:23, 08:21 - 09:33, or 09:40 - 11:11.
- Bus 432 or 433 Norrköping – Kolmården. For example, 10:00 - 10:09, or 11:47 - 12:00.
Kolmården Djurpark is 27 km from the Norrköping central station. Waiting times between train and bus appear rather long (more than 30 minutes). If you're many, you could consider taking a taxi. When booked via SJ, a taxi costs roughly 600 SEK, but a look at the prices for Taxikurir Norrköping suggests it's probably less than 400 SEK there (I get 330 SEK based on their prices). This can be compared to the bus, which costs 74 SEK for an adult and 51 SEK for youth or senior; so for a family of 2 parents, 2 children, the bus would be 250 SEK.