Make sure you wash out the old gunk before regreasing. I usually use a spray called brake cleaner/degreaser or something like that. It foams and smells good
It leaves all parts completely free from dirt and grease without attacking paint or plastics.
For regreasing I use my trusty old Mobilgrease HP (aka XHP 222) which will not separate, run, harden, dry out or anything, but there are lots of other quality greases that work well.