How to consume external REST services from within WordPress

As shortly discussed with @Jose in the last Coding chat session, I would like to consume REST API services from within WordPress.

This is the API I am looking for: Expedia EAN (http://developer.ean.com/docs/room-avail).

I would like to set the parameters of the REST call based on user settings, e.g. taking the dates from a calendar control.

The results from the call should then be parsed and be displayed in a table.

As I am new to WordPress developing I would appreciate any advice I can get from you, e.g. are there plugins which could to the heavy lifting or where should I look for a starting point regarding development.

Thank you!