Splurge. Get the year long pass for the family at the zoo or the museum or both.

It takes so much stress away. Once you've committed to the year long pass, you've helped out the zoo/museum, but you've also helped yourself out.

You no longer need to worry about maximizing your trip.  You won't feel pressured to see everything during your visit. You can go at a relaxed pace - at your child's pace - and enjoy things with them.  Instead of rushing them from one exhibit to another to "get your money's worth".

Each trip you go on with your child will be different. They will have changed. They will have new interests, or renewed interests.  You'll be able to encourage them all. If they want to stare at a fountain (or even just a drinking fountain) for 20 minutes on the trip, you won't mind, knowing you have prepaid for the year.

You'll always have an activity for the slow days. You can plan out trips with them. Discuss the things you saw - building their memory and story telling skills. Discuss things you will see again - to work on their imagination - wonder what the elephant is doing right now?

Most places in the US will credit you back your day entry fee if you upgrade to a year pass the same day. You can do that to see if the zoo/museum is interesting enough for your child for the year. Odds are, it will be. Enjoy it - be a kid again - see it through their eyes together, and you'll have a blast. 

Learning together will create a bond that'll last a lifetime.