You can easily get to the San Diego Zoo in just over 90 minutes from Anaheim on a Saturday morning. Especially if you left by 9:00am, the traffic on I-5 should not be a problem. The fast lanes of that freeway will be zipping along at 75 mph, especially through Camp Pendleton, but even at 65mph you'll get there in about 90 minutes. (There's a great Starbucks I frequent 10 blocks down Katella Ave. from Disneyland, directly across from Anaheim Stadium, if that would help get the morning going better. You can then jump on the 57 Orange Freeway southbound, which automatically dumps you into the 5 Freeway southbound about a mile past the stadium.)
The San Diego Zoo is very popular, it should be busy on a Saturday, and I would give yourself about five or six hours there to see and do most of it. It's a physically large zoo, and it has several new exhibits that are massive and can take 30 minutes or more to see. Take advantage of the one hour long zoo bus tour they offer, and definitely sit on the upper level of the double decker bus. The bus tour requires a more expensive ticket, but it's worth it and very informative. The Skyfari ride is also great, and reminds you of the old Skyway that Disneyland and WDW's Magic Kingdom used to have. And the views of downtown San Diego are wonderful.
Petco Park in downtown San Diego is about a 10 minute drive down the hill from the San Diego Zoo. You can just take surface streets, and I'm sure you'll Mapquest that short drive from Balboa Park. The Gaslamp District around the ballpark is a very popular place, and on a Saturday night it will be very busy with locals and tourists alike. But as long as you don't mind paying for a big parking tab you should be fine.
The good news is that you've picked a fantastic time of year to make the trip. The weather in May is always stunningly perfect in Orange County and San Diego, and the summer crowds won't have descended yet. Be sure to bring lots of sun screen and hats if you're from a northern clime though. The golden California sun is approaching it's most intense in late spring, and sunburns can happen quickly that time of year. But also bring a sweater for night time, as the beach fog will roll in after sundown in downtown San Diego and it will cool off a bit.
