Panini's can definatly be made at home.
When I was about 13, I insisted on owning a George Foreman grill. So, I found the small size at kohls for 20 dollars. Needless to say, IMO, I find it absolutley useless for cooking anything except paninis and hot dogs.
To make a panini at home, the one thing that shouljd remain constant is the bread. A nice hearty bread, like a ciabatta loaf, or a boule, pretty much any type of bakery bread, just not sliced bread. the contents can be whatever you choose. Cold cuts, leftover chicken, turkey, cheeses whatever you like.
to turn it into a panini...
If you have a stove top grill pan, just heat it up, place the sanwhcih on it, and just put a skillet on top of the sandwhich, after a few minutes flip over the sandwhich and put the skillet back on top.
If you dont have a grill pan, you could do this on the outdoor grill I suppose, or even in a regular fry pan, tho it will have a different texture then a panini.
Personally, I make them on my George foreman machine, tho I did need to "break" the hinge so that you could lift the top up and over the sandwhich, otherwise it didnt fall flat on the bread.