My boyfriend is from USA and I am from UK. We are wondering which is the best and most straightforward method of being together, i.e should he move to UK or should I move to USA. Also how long do each of us have in the other country, and, as he has proposed to me, is it easier to immigrate as a married couple to either UK or USA? I am planning a trip within the next couple of months so would appreciate any help or information. Thanks.

With a UK passport you can get a 90 day US Visa Waiver Program entry or you can apply for a US B-2 Tourist Visa which will give you up to 6 months.
With a US passport he will normally get 6 months in the UK.
In either case the non-citizen CANNOT work during their visit, you will also find that while you may consider being with the boy/girl friend ia great reason for visiting the other country do not expect the immigration officer to share your enthusiasm as they know from experience that hormones trump immigration law even in otherwise law-abiding people, it might be wiser to present it as a sightseeing visit.

Thanks Mark, I had heard of the B-2 Visa but wasn't sure if it still existed or how long it takes to get etc. Have you any further information about that? Yes I would be going as a tourist, I previously lived in the USA and both my children were born there, although my ex husband (their father) was English, and we were living there for his work (Michelin).
If you come to the US on a fiancée visa, you must marry within 90 days of arrival. However, getting the visa can be long & complicated. Although you can visit the US while waiting approval of the fiancée or spouse visa (for example under the Visa Waiver Program), you may be held up to higher scrutiny to prove it is a temporary visit.
If you enter the US under the VWP or a B-2 tourist visa, with the clear intention of marrying and not leaving the US after the wedding (whether or not you have applied for a fiancé visa), you are guilty of visa fraud.
Many people find that an immigration attorney is helpful in getting through the visa process.
Expect the processing time for either visa to take several months, with or without an attorney.
What to Expect When Sponsoring a Fiancé or Spouse for a Green Card from Nolo Press, which specializes in legal issues. They sell a handbook about the process. If you don't want the eBook, Amazon.co.uk should have it; be sure to get the most recent one.

To get a B-2 visa you have to make the application then schedule an interview, the waiting time for an interview varies by the time of year but expect perhaps a month, if they accept the application the visa will be issued in a week or so, BUT with a UK passport you are eligible for a 90 day VWP entry without all the visa hassle so they are gong to want to know why that is not adequate, you had better have a good answer prepared.

Visa appointment wait times.
Right now, London is showing 24 days for visitor (B-2) visas, and 23 days for spouse/fiance/fiancee visas.

OK so maybe go for the VWP in the first instance, then have my boyfriend return with me to UK then take things from there. I mean we are both legit in our relationship, neither of us has bankfuls of money, although I can support myself while I am in USA. Maybe an immigration lawyer is the way forward.