There's Neot Kedumim, outside Modi'in, which is accessed by bus from Modi'in
and Tel Aviv (possibly also from J'lem directly).
J'lem Biblical Zoo is a great day out.
You can take the trains up and down the coast for beaches.
There's techellet tours in Zichron Yaakov, and ZY is a lovely artsy place to
spend the remainder of the day.
Alternatively, you can hire a guide to take you to interesting sites. I can
recommend www.danny-the-digger.com for archaeological or otherwise