Using Google Maps with the public transit option selected, or using CTA's trip planner can be great resources for figuring out how to get from anywhere in the city to the Lincoln Park Zoo using public transit.
On the L
The nearest L stop is Armitage, near Sheffield Ave, which is served by both the purple and brown lines. From there, you can either walk to the zoo (a little over 1 mile, or roughly 20 minutes), or take bus 73 east on Armitage to the stop at Armitage and Clark, which is a short 4 minute walk to the Brauer gate at the zoo.
By Bus
There are many buses that serve the Lincoln Park Zoo. From the loop, these include
bus 22 from the Loop up Dearborn and Clark,
bus 36 from the south end of the Loop and going up Dearborn, State, and Clark,
bus 151 from Union Station up Michigan Avenue, and
bus 156 from Union Station and the Loop, going up LaSalle. Buses 22 and 36 continue a good distance up for those coming from the north. Of those buses, only bus 156 doesn't run on the weekends. All four stop at the West Gate which is the entrance I generally recommend from the west (the closest stop is Clark and Webster for 22 and 36, and Stockton and Webster for 151 and 156). Alternatively, 22 and 36 stop at Clark and Armitage while 151 and 156 stop at Stockton and Armitage, both of those stops are close to the Brauer Gate and closer to the Farm In the Zoo.
If you're coming from the west, bus 74 along Fullerton and bus 73 along Armitage will get you within a couple of blocks of the zoo. Both of those buses run on the weekends as well as weekdays.