You definitely can do it. You just go to the family court and get a D8 form, and file it. You file it, and if it's unopposed (which obviously it would be) you get the Decree Nisi a few weeks later, which means that you are divorced. Most divorces have three parts- that bit, the financial arrangements, and arrangements for the children, if any. You don't have to make any financial or children arrangements through the court if you don't want to, but it might be an idea to apply for a "consent order" which means that the court rubber-stamps whatever you have agreed between yourselves. This doesn't cost much, and doesn't need a solicitor. If you do that it's harder for one person to change his or her mind and demand money or more/less time with the children later on.
One unfortunate bit is that however amicable the divorce, you have to put grounds for the divorce into the petition, so one person is going to feel accused- be sensitive about it; you may both know it's only a formality but it may not feel like that when it's in black and white.
But you definitely can do it, and since you have to agree on everything one way or another, in the end, it's a good way to start. Family court judges love couples who aren't rowing, so they'll be very happy to help you and de-mystify the process, well, they certainly should.