Things that pop to mind... in the eBay thread someone mentioned that a gallery ships items wrapped in several layers (3) of bubblewrap, then inside a shipping box with styrofoam peanuts - very good advice.
For customs - if you are selling the painting and need to declare value as merchandise (like I do) just put the price of the painting - do NOT include the shipping value. I had someone do this to me once. argh. Seems obvious.
Otherwise, if it is a gift or to a personal friend, etc, tick the "GIFT" box. You'll still need to put a value, but gifts are customs-free up to around $100-200 I think. Do NOT put any correspondance in the package, having a letter inside can make the price go up. You can put in a receipt or business card though.
I highly recommend you buy insurance for the items. It's usually very cheap and handy just in case the package is damaged or goes missing. My policy is that they can have their money back or commission a new piece with the insurance money.
Personally, I find FedEx the easiest way of shipping overseas and use it for larger items and they're so helpful! (or UPS) With UPS though, they tend to leave packages on doorsteps in the US so you may need to see if they have an option requiring a signature. I've never had a UPS delivery here, although FedEx requires a signature.
You ask for quickest and cheapest and those 2 words don't go together I'm afraid.
Quickest normal mail is airmail. Cheapest is surface mail, which takes 8-12 weeks. As I ship London-Chicago very often, yes, it really does take that long! Airmail takes less than a week, usually (yes, really! packages from Chicago usually arrive here 3-5 days after my mother sends them!). Also, do NOT trust prices on the USPS.com website - I had several problems with them this past winter when I priced things online then took them in to send.
Hope that helps a bit! I don't think there's any particular pitfalls... I've never done anything special shipping paintings Chicago to the UK.