You can use caulk to fill holes left in the siding once the gutters and downspouts are removed.
How to take off rain gutters.
This will make removing the rest of the gutter easier.
If the gutter is very long be cautious removing it.
This is usually done by removing screws or using a pry bar to remove nails.
How to remove a piece of gutter.
This is as simple as removing each gutter hanger or other device that holds the gutter in place.
Gutter nails or spikes used to hold rain gutters in place can work themselves loose with seasonal movement.
Then simply place the block in the gutter and use your hammer or pry bar against it to pull out the nail and remove the ferrule.
The rain gutters can be attached using nails screws or rivets and some type of strapping or braces to hold them in place.
Measure the gutter and cut the block to fit snugly between the fascia and the outer lip of the gutter filling the void with solid wood.
With the above safety procedures in mind use a power drill in reverse or a screwdriver to remove the screws underneath the roof or in the siding holding the gutter in place.
You can make a pry block from scrap 2 4 or 2 6.
Remove the gutter once all hangers are gone.
Your next step will be to remove the downspout.