I would imagine straightforward means that your employer has done the appropriate homework to show that they've tried seeking help within the UK/EU and that you've got enough qualifications to show that you're more qualified than everyone in the UK/EU. If you're transferring companies, than you need to have adequate evidence of time with the company, etc.
You can travel while waiting for the WP. Once you get your WP, you need an entry clearance visa to come here, which is in your passport,and you apply for in the states.
My WP was done in October and it took several weeks, 6 or 7ish I think, I wasn't able to apply for Entry Clearance until December. Mine application was straightforward I think...