I recently took a 7 days trip to the UK and Ireland and this blog is basically to share my itinerary and do's & don'ts of the trip. If you're looking for some help around planning a similar trip, hope it comes handy!
First things first, how to apply for visa and book flight tickets?
You will have to apply for a visitor visa for traveling to the UK or Ireland. You get a combined visa under BIVS (The British-Irish Visa Scheme) in India, and you will have to apply for visa for the country you're landing in. If you plan to land in UK (I took a flight from Delhi to Manchester), you can go to https://www.gov.uk/apply-uk-visa?utm_source=visa-application-form&utm_medium=internal-link&utm_campaign=svv-apply-visa. You don't need an agency to get this done - the application process is pretty much straightforward!
As for the flight tickets, the usual skyscanner & makemytrip drill works just fine! Try taking a flight from Delhi, and book yourself a domestic return ticket to Delhi in case you're living somewhere else. Worked for me pretty well!
Also, do carry AT LEAST 100-120 pounds in cash, because you can only pay in cash if you wish to travel by bus inside the city.