Egypt at it's core is two landscapes. The lush green strip of land land around the Nile. And the desert. At least nowadays. Going to Wadi Al Hitan, will take you outside the green strip inside the desert. But then, there they are: the whales. The explanation is that millions of years ago (40?), this was a mangrove forest.
It's a bit surreal and a very unique visit, as it combines a hike in the Sahara desert with stunning fossils. It feels almost like an art installation.
I had stayed the night in Tunis and joined a group of Egyptians (women) on a desert tour. We had plenty of time at Wadi Al-Hitan. The site has an onsite museum which I skipped and instead spent more time with the site proper. Most Egyptian visitors seem to stop fairly early on, but I kept going. Eventually, doubts settled in how far to go as I was all alone in the desert. But I think I made it to the end of the trail before returning. As pointed out by Els, you have to walk back the same way you came.
And one more comment: My mind seems to be off, but I didn't see mushrooms in the stones. ;)
Getting There
The starting point for most excursions to Wadi Al-Hitan is Tunis. Easiest option is to take a minibus from Giza to Tunis. They go right up to the park entry in Tunis with the option of getting off before.
If you are coming up the Nile from Amarna/Luxor, you can get off at Bani Suwaif. If possible, have your hotel prearrange a taxi for you. Otherwise, be prepared to spend an hour or more while Egyptian police arrange a taxi to Tunis for you. Sit down...
Tours can be arranged online. Other visitors listed very high prices for a visit. I think prices can be brought down significantly, if you stay a night in Tunis and don't try to rush this as a day trip from Cairo. My hotel had the cheapest quote.
If you want to drive yourself, I think the road to the core zone is okayish enough. A jeep is not needed. But to get to see more parts of the site, you should get a 4wd.
Last but not least, on security. When we passed security, the guards were initially fooled by my darker looks. As soon as they checked my EU passport, they were about to give us an escort, but eventually let us go on our own.
While You Are There
The desert around Wadi Al-Hitan is beautiful and really popular with Egyptian tourists. Tunis and the Qaran Lake also offer more sites to explore.