If you change your display name, all of your old questions and answers will use your new display name. However, if anyone has @replied you in comments, the @replies will still use your old display name because they are part of the actual text of the comments.
The old display name will also continue to appear on Community Wiki posts and chat transcripts.
It's important to keep in mind that your old display name will always be visible from websites that scrape and archive old content, like the Wayback Machine.
There is a "Recent Names" section on your profile that keeps a log of your recent display names, but it is visible to only you and moderators.
For the most part, the transition from one display name to another is relatively seamless, but there are a couple of downsides, the first being the lingering display names mentioned above, which can make discussions/edit histories confusing for future visitors.
Another is that you lose any to all reputation (not the points, actual reputation) associated with you by other users. To put it more simply: people won't know who you are, so you lose all association and trust associated with your display name.
Of course, your name now is user237730, which doesn't carry much, so changing it can't hurt, but it's something to keep in mind for later if you change it to something more substantial.