Board the Eurostar at London St Pancras, and make a change at Brussels-Midi for the high speed Thalys train to Amsterdam. This leg of the journey should take just under 3 hours. Then, in Amsterdam, you'll be able to hop on an InterCity train to Utrecht; these services run every 15 minutes with a journey time of 30 minutes.
History buffs will love uncovering Utrecht, the fourth largest city in the Netherlands, which can trace its foundations to as far back as 76 AD. Thanks to its large student population, Utrecht is a great place to visit if you're looking to enjoy a few nights out on the town. Janskerkhof Square and Oudkerkhof Street are both good destinations for those seeking the most lively bars and clubs.
Here are some other resources that might have the information you need