Hiya
I did it the other way an imported a Ford F150 from the US to the UK, but generally speaking, you'll be able to ship from here to the US without too much fuss. Pricing wise, I think you'll pay approximately the same US>UK and UK>US, things like shipping in a open container, sealed container, amount of insurance, timings, handling etc is what will make the difference.
We shipped the F150 from Texas, it cost us £1250 (ish) in an open container, I think a sealed one was at least about £200 more, seeing as we had full insurance etc, we just opted for the cheaper option, subsequently, we 'should' have gone for sealed as the truck was Filthy when it arrived but thankfully completely undamaged which according to sources is quite rare!
From what you've said, are you permanently taking the car out of the UK? if so, you'll need to inform DVLA and complete the appropriate paperwork etc. Also, are you planiing on keeping it in the USA? if so, you'll need to find out if it's US road spec legal - and from my enquiries, the US is a bit backwards when it comes to importing cars into it's borders - a lot of red tape and stupid requirements basically, if your Honda doesn't meet the criteria, then it's pretty much pointless taking it over. You need to take into account making the car US road legal (headlamp beams and patterns, rear lights & indicators/blinkers, bumpers etc)
If you are just doing a temprorary thing, then you'll need to do the basic stuff above and check for individual states you'll drive through local requirements. Check that the car can also run on much lower grade unleaded petrol - UK has a minimum of 95 RON octane petrol and the US I think it's lowest grade is 89 RON and the highest is 93 RON - Octane boosters may help - otherwise if you're car cannot handle lower grades, it'll be pointless to take it.
there's probably more, but the above is most of it-
Hope it helps!
Dennis! West London & Slough UK!