Stump grinding costs vary greatly, but none come cheap, you need to get some quotes, the machines are quite big so check if you have enough room for access?
Digging it out really depends on how old/big it is re costs etc again you need to get quotes and dont forget to include removal from the site?
My stumps were taken down to earth level by a chain added to the chain saw by the tree surgeon and stump killer applied.
Stump killer which is applied at time of cutting, it does kill the stump, But does not get rid of it, you then have to wait some years for it to rot down.

Contact tree surgeons in your area - they've usually got someone on their team who does the stump grinding and removal, and get two or three quotes. Prices vary between companies, and from region to region in the UK.

You have me worried now!- I just planted a fruit cherry this season.....I didn't think they were that troublesome!

Well if you've only just planted it, Begoniamatt, hopefully it'll be after you're long gone before it needs the remains removed!

And that advice relates to all trees, not just cherries! So don't give up planting trees will you?