For many young adults of means in Edwardian England, an extended trip to Europe was considered an important part of completing their education. It combined exposure to the culture and history required of all refined individuals with the excitement of travel to exotic locations, even if that travel was accomplished with all the comforts of home in carefully selected lodgings and travel companions. So, in one sense, the journey was the capstone of Lucy's education as a proper young lady.
In a deeper sense, the trip to Italy became Lucy's education in life and who she truly was. Lucy learned through her encounters with new places and new people, most of all George Emerson, that she did not have to yield her personality to the rigid expectations of the society of that time. Lucy traveled from being a docile follower of directions given by others to being “a rebel who desired, not a wider dwelling-room, but equality beside the man she loved” - a very significant journey, indeed!
The answer above is an interesting one and represents one valid reading of the novel. However, there is an alternative reading, one supported by Forster's later appendix to the novel, 'A View without a Room'. This appendix can be found in some of the Penguin Modern Classics editions of the text and online (I have included a link to a copy of the text below for ease of reference). This revisiting of the characters, 50 years after the original publication, shows that Lucy's and George's marriage was not intended to be a happy one and questions the transformative nature of their journey and their escape to their own shared room and view at the end of the novel, the more conventional reading of the novel's closure and one pursued by many critics, from PN Furbank, Forster's most famous biographer, to modern studies such as Frank Kermode's last critical work.
The alternative reading highlights the circular nature of Lucy's and George's journey, returning to the Pension Bertolini at the end of the novel from where they started at its beginning, both dominated by the 'view' of the world that they had received during their upbringings. This return to the Bertolini might suggest that they have gone 'full circle' and for all of their struggles, might not have learned anything at all, rather that they have similarly chosen to submit to one 'view' which is forced upon them both. This is perhaps made all the more pertinent when one considers the 'room' that they end up in - if one reads the final chapter of the novel carefully it becomes clear that the room they return to is not George's but his father's room. This seems to be of a piece with the struggle which Mr Emerson articulates for George at the start of the novel, that 'the things of the universe don't fit' - he does not seem to be able to easily reconcile himself to his father's socialist view of the world while Lucy similarly cannot wholly reconcile herself to the bourgeois, socially conservative view of the world that her family inhabit at Windy Corner. However, it is notable from the resolution of Chapter One that, when Charlotte Bartlett finds the 'enormous note of interrogation', the question mark that represents George Emerson's questioning of his father's perspective, that she has exchanged her room for George's and thus Lucy inhabit's the father's room and with it, Mr Emerson's 'view'. Thus, when they return to the Pension Bertolini at the end of the novel, it is to return to Mr Emerson's room and his view once more, the socialist 'view' which George has also now accepted and which, in the penultimate chapter of the novel, Lucy is finally won over to. In acceding to Mr Emerson's 'view', Lucy becomes distanced from the 'view' of the world espoused at Windy Corner. This isolation and the unhappiness that follows is summarised in the later 'A View Without a Room'. The later essay is the unusual step of a writer who appears so frustrated with the misreading of one of his first novels (which, although published in 1908, was one of the first things he wrote, an original version having been produced in 1901) that he was moved to correct the misreading 50 years later, questioning the ending of the journey as being the 'happy ending' of redemption and happy marriage which others have assumed it to be. The rather gloomier reading of the novel does place A Room with a View in a more obvious connection to the rather less optimistic endings of all of Forster's other novels which, Maurice apart, include less happy ends to romantic relationships.