What are the best UX practices for integrating a website with a call center?

You can of course use analytics that will give you statistics, but will not answer "Why?". Gathering Customer Feedback is crucial and You have to be preprared for low response rates. For better understanding of Customers' interactions with Your site You can use tracking/heatmap solutions (i.e. Crazy Egg or Hotjar). But most important is to define goals and identify steps that lead to them - what do You expect Customer will do or act like? What needs to be done to acheive it? Then start monitor those steps, gather feedback, mae changes and see if this works looking at stats.

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I owned an SMB in the health & fitness space and spent several years running a SaaS/PaaS startup in the business productivity space. There are lots of WP developers out there, I'd start by looking at elance. Have you done mockups so the developer can accurately quote the work? It's worth a little design expense up front to make sure your vision is communicated clearly. I'd also make very certain that an off the shelf solution won't work. Zenplanner and Mindbodyonline are two cloud-based CRM/studio management solutions tailored for fitness businesses, I assume you've ruled those out. There are many generic SaaS CRM solutions out there as well so I'd really make sure that the "ideal" experience is really that much better than "good enough". I used MBOnline and it was not perfect but it certainly didn't hold back my business. I'd also recommend finding someone else in your space who has what you want for customer experience, and just ask what they use. Assuming they are not a competitor they will probably be forthcoming with the info. Happy to jump on a call if you'd like to discuss further.

Solidify your idea by making sure that there is a market there. That means putting everything into a presentation, similarly to create a Product specific business plan. Speak with your tech team to decide upon technologies and commercial of the shelf software licenses, including hosting. Design a SIMPLE and good looking UI/UX and apply standard software methodologies such as SCRUM from development, Monitor progress and milestones. Most importantly target on "small" and doable product feature. One by one start adding. Place your product in the market and test-bed it fr FREE. Evolve and change features and functionalities. Remember Marketing is NOT everything ... but it is!

This would really depend on your goal of the lead channel. If you want to deal with warm leads give http://inspirebeats.com a try. InspireBeats will help you find and build a list of targeted leads, do research on them, and warm them up for you through a personalized outreach. That way you have an additional channel of leads flowing in for your sales reps.