I am going to be travelling in EU soon, mostly Switzerland, and the neighbouring countries, but also a few others, and was wondering if my Provisional UK Driving License is accepted as ID, or would I have to take my Passport with me everywhere?
No, you can't use a driving license, because what is required is not just a proof of identity but a proof of nationality. You could have a driving license if you were a British resident, and it wouldn't give the right to enter Switzerland without a visa. Generally speaking, you need a national identity card or a passport.
You can consult the Swiss list of accepted travel documents. Assuming that you're a British citizen, you need a passport (or a seaman's book). A British passport without British citizenship also lets you in without a visa, for stays up to 90 days. Beware that
British Identity Cards containing under the heading «nationality» the entry «British Citizen» (B.C.) are not valid for entry into Switzerland anymore.
You certainly don't need to take your passport everywhere. Whether you need to have an ID on you at all times and what is accepted depends on the country. In most cases, being able to produce your passport the next day should be OK (but there are exceptions like the Netherlands where not having a recognized ID on you makes you liable for a fine, no matter what your status is).
In countries like Germany or France, your driver's license (provisional or not) usually would not count as an official ID but should be OK to satisfy the police in the unlikely event that there is a control (here some details about France). What could happen if the police is not satisfied or somehow wants to make your life miserable is that you have to follow them to the police station but not having your passport on you is not an offense in itself.
In Switzerland, Swiss citizens don't have to carry ID but I don't know about foreigners. You will also need your passport to be able to take the plane or ferry to and from the UK.