I'd probably start with a base charge of $10-15 per visit if it's a home not far from you and the pet is easy. I'd adjust the rate depending on the number and difficultly of the pets. Obviously, even though the number of pets is the same, a home with 3 guinea pigs is less work than a home with 3 dogs who are all on medication and need to be leash walked around the block with each visit.

I'd probably charge 3x your per visit rate for staying over night. So if your per visit rate for a client would work out to $15/visit I'd charge $45 for staying over night. During an over night stay you'd be doing the work of 2 visits, at least (an evening visit and a morning visit) plus you'd be staying there all night.