Twycross Zoo
The Zoo website doesn't give any indication of how to reach them by public transport, but there is a bus each way, the 7, from nearby Atherstone (which itself is one stop from Nuneaton on the train). It only seems to stop at the Zoo itself once in the morning and once in the afternoon to return to Atherstone. The morning bus leaves at 10.03 from Atherstone bus station, or 10.07 from outside the train station (it's not that easy to find, and isn't numbered on the sign, but it's at the end of the road that turns right towards the Co-op Food store, on the side of the road going away from the Co-op). The return bus leaves the Zoo at 15.29 from the same space they drop you at in the morning.
The bustimes website linked below has the option to view each stop on a map, which is handy to both find the actual bus stop and because some buses show live-locations on the map. The bus doesn't run on Sundays (or bank holidays it seems)
7 - Measham - Twycross - Atherstone - Fenny Drayton