Requirement for Photo ID
Generally, you will find that the requirement for photo identification is fairly limited. I think you may find the hassle of using something other than your passport isn't worthwhile, unless you are young enough to need to prove your age when buying alcohol or tobacco, or have specific work-related access-control requirements.
In almost all cases that I've been asked for photo ID, it's been a organised situation - like applying for a bank account or similar. The only exceptions to this have been:
- When driving, where my driver's license was required.
- When visiting a secure datacenter which had a policy of requiring photo ID that I wasn't aware of.
The most common form of photo identification is almost certainly a driver's license. If you have an EU driver's license card, you will almost certainly find it's universally accepted as proof of identity in the UK, since the UK card adheres to the EU standard.
In the unlikely event that you run into trouble with an EU license, you could convert your EU driver's license to a UK driver's license as per https://www.gov.uk/exchange-foreign-driving-licence for a fee of £50. There are certain limitations to when you can do that, such as how long you've been in the UK.
Biometric Residence Permit
Depending on your visa / residence status, you may find that you are required to register for a Biometric Residence Permit. This is a legally acceptable form of photo identity, but is only applicable in certain circumstances (such as when getting Indefinite Leave to Remain in the UK). It is thus quite unusual, and you may only be required to get it years after you arrive.
Proof of Identity in the UK
Proof of identity in the UK is politically and practically complicated. Overall there is no universal method for proving identity in the UK, as many claim that introducing such a structure will lead to a loss of civil liberties. I cannot link to the page (due to my low ranking) but a search for the No2ID campaign will explain more about this.
Identity is thus normally proved by having letters or utility bills that are destined for your name and address. There is no requirement for those born in the UK to have photographic proof of identity for almost any purpose (including voting or even buying a house). Passports are not universally held, but are required for leaving the country - I would guess that they are the second most frequent form of photo identification.
Overall, the lack of universal photo identification means that the requirement for photographic identification on a day-by-day basis is very limited. It is only really required when doing something like opening a bank account as a foreign national, or registering to study or similar. You may find it easiest to stick to using your passport for these reasonably rare circumstances.
If you are running into problems buying alcohol or tobacco products, a private company produces a card that retailers will accept for this specific purpose at http://www.pass-scheme.org.uk/ Note that it's not proof of identity so far as I know, and I think it's very unlikely your bank will accept it.