The best time for a walk through conero park is definitly spring, when it's not crowded yet and nature starts showing its colours.MOst organised excursions though start from latre may or even june as before, there is not enough request, but, if your porupouse is just a walk in the nature and some photography, then yoi can easily do that on your own.
There are several ways to explore Conero park, probably the best thing you can do, is going to local information office and ask for routes and eventual planned guided tours, usually in summer there are weekly interesting tours.
If you are in Ancona only for a few hours, just take a look of the city, there are some nice monuments and churches to visit, one example can be Saint Ciriaco church, but, if you can stay a few days, one can be enough, do not miss the Conero mountain, there you can find some great views on the seaside.
The views you get while walking on the Monte Conero.
Conero has some really great small beaches, try for example St. Michele beach, Conchiglia, Portonovo, Mezzavalle, Sassi neri, Urbani.
Le due sorelle beach(two sisters), is a fascinating small beach in front of two rocks(the two sisters) on the sea.