We feed them suet cakes - available at animal supply places (like Agway), Target/WalMart, Christmas Tree Shop and Ocean State Job Lot. They are from 79-99 cents at the last 2, and up to $1.79 in the supermarket. You can hang them in a special wire feeder - not expensive - and then hang it from a bird feeder or clothesline, whatever you have. Don't hang it from a tree branch too close to a tree trunk or the squirrels will eat it all. If you have to hang it from a tree branch, invest in a squirrel baffle for the top of it. You can also use a big aluminum pie pan - punch a hole in the center, then thread the chain from the feeder up thru the hole and up the long chain or twine (whatever you are using) up to the tree. Otherwise, use a birdfeeder pole or one of those shepherd's crooks you can find in garden supply stores - something freestanding away from any structures that give the squirrels access!
Store the suet cakes in your house (especially the refrigerator - see below) but not in the garage - the squirrels will tear them apart!
The woodpeckers are looking for insects in the wood of you house. If you have anything in there, they can tell! If anything takes up residence behind a wooden shutter or a wooden clapboard, the woodpecker tries to get to it the only way he knows how.
A suet feeder gives them ready access to what they need - it doesn't matter if it's suet with peanut butter, berries, or fruit - woodpeckers are meat eaters so they are looking for bugs or something else meaty - the suet gives them that.
To make handling suet easier, refrigerator the little package - it pops out more easily and slides into the feeder without getting soft all over your hands. That's going to be especially helpful as the weather warms up.
Sometimes you can buy organic/natural repellent for some critters which is made from the urine of something else that is a predator. I know you can buy something to deter the chipmunks that is made from fox urine, for example. Maybe you can find something similar for woodpeckers - not sure what it would be - maybe something from an owl or a hawk??? You could google to find out.
If you can keep the birds off your house, they are really beautiful to watch and enjoy!