Hi James.
Sunburn should not be an issue in April unless you are very pale, and I wouldn't worry about spiders or snakes. You probably won't see any.
The coastal scenery north and south of Sydney is beautiful -- some of the best beaches in the world -- but if you want a uniquely Australian experience you'll need to head west and I recommend that you include some inland flying in your travels.
You may wish to allow yourself more time and also consider something a little faster than a C152 if you have the resources.
Send me a private message if you have other questions. There's a slight possibility that I could accompany you around that time.