What is the best way to remove any sediment build up from having hard water from a washing machine?
Pretty much anything acidic will do the trick. There is an overview of citric acid-based products you can purchase, and home remedies such as lemon juice and vinegar. Here are more details on using vinegar.
There's a product called CLR (Calcium,Lime, Rust) that you can pour on a sponge and scrub down the drum. Use gloves! It probably wouldn't hurt to pour the rest of the bottle in and run a small load (no clothes) to clean out the piping as well.