One of the best-priced and best-located places to stay in London is Carr-Saunders Hall. It's around the corner from Euston Station and Tottenham Court Road and handy for Goodge Street underground. It's the halls of residence for the LSE - the drawback is that you can only stay there during the university holidays. But it does make London a little more affordable.
Theres alot to . ..a lot if planning....I have almost a year to plan...whichbit might take that long...hiw about this, if I make london my base and travel from there returning after each trip will this make my travels easier+
You do not NEED a long time to plan but you are doing the right thing & starting early. As a US citizen you do not need a visa to enter Schengen area countries or UK or Ireland or most non-Schengen countries so obtaining visas is not a TIME problem.
NOTE - Entering the UK on this page https://www.gov.uk/check-uk-visa/y/usa/tourism also says this -
>>>> You won’t need a visa to come to the UK
However, you should bring the same documents you’d need to apply for a visa, to show to officers at the UK border. https://www.gov.uk/government/publications/visitor-visa-guide-to-supporting-documents
You may want to apply for a visa if you have a criminal record or you’ve previously been refused entry into the UK. <<<
You could take a lot of time planning where you want to go, & what you want to do. I like to plan but you might like to just wing-it.
Using London makes no sense to me as a base to visit continental Europe, but an excellent, if expensive, base for the UK.
You could totally wing-it - just get on a flight tomorrow to say France ( part of the Schengen ) wander around the Schengen area for the maximum 90 days. Go to UK for another the 180 days you are allowed visa free in UK . Go to any Schengen country & wander around for another 90 days. That is your year & there are still non Schengen European countries to visit, which are probably cheaper to visit too.
Hardly any planning needed until you wake up in the morning - just remember to abide by the rules of allowable days.
You will not be allowed to work though & you may need to show proof that you are able to support yourself for the amount of time you ask to stay.
On the other hand, you could spend the rest of the year trying to work out where to go & how to get from A to B.
I suggest you start with the 'Planner' & 'Map' here on Travels point and get a feel of how it all fits together. Come back to the forum to ask specific questions - try not to be vague with your questions. Use Google search & your State Department.
If you are wanting to work while away you will need visas, & you should check the various Working Holiday Visa rules for the country you want to visit - More research of course - it never ends.