Yes, it's definitely travelling to "tomorrow".
It's even conceptually possible, in one very special case, to travel
While the technolology to achieve this exists, it's not readily available and would cost tens of millions of dollars to do it. Details at end.
Our time zone system involves "tomorrow today" as a very real part of reality.
When you phone your destination and find that it's Saturday there and Friday where you are and they are closed for the weekend, it's a very real 'tomorrow' that you are experiencing.
Similarly, if a booking or a competition entry or similar closes at 5pm on Friday in New Zealand, Then the fact that it is still Thursday in San Diego when the deadline is reached will not excuse a missed deadline.
The difference is so real that last year Samoa decided to jump ahead a whole day to match up with New Zealand. Samoa had been operating exactly one day behind NZ but the large community of interest between the two countries meant that people were being substantially disadvantaged by the 3 business days a week that they shared. ie Samoa Friday was NZ Saturday, so the NZ end business may well be closed. And NZ Monday was NZ Sunday so the Samoa end business may well be closed. This is especially likely as Samoans tend to observe Sundays as a day of worship and rest much more closely than most other countries do.
All days are 48+ hours long!
An important factor which influences this situation and which very few people are aware of is that all days on earth last for at least 48 hours and often are 49 hours long. This is before you start to wonder about eg Samoa and Tonga which warp arrangements further.
The 48 hour situation occurs because, when eg NZ is just finishing a 24 hour day at say 23:59, immediately to the East across the dateline the day which is just ending for NZ is just about to start for eg Western Alaska (or would be if it used the time zone it "should" but it doesn't - more on that below). As NZ ends its 24 hour day Alaska starts the same day and finishes it 24 hours later, so the day exists somewhere on earth for 48 hours. However, in Summer NZ adds + 1 hour of daylight saving time and so "begins the day" one hour (one time zone) sooner than otherwise so a given day exists at some location or other for 49 hours.
So, effectively, there are two whole 24 hour days in existence at once, spaced 24 hours apart in time. To complicate or simplify things (choose one) no populated place anywhere has chosen to use UTC-12 = the very end of the day on earth. Western Alaska which is geographically in UTC - 12 (12 hours behind GMT and 24 or 25 hours behind NZ) instead actually uses UTC-9, which puts it "next door to" (one hour behind) Pacific Standard Time - the whole US West coast.
Fortunately for fun there ARE two places on earth that do notionally use UTC-12. As they are both uninhabited nobody usually notices. Baker Island and Howland Island in mid Pacific are both in the UTC-12 time zone. Their Monday ends 48 or 49 hours after the same Monday begins in NZ.
Better still, In Southern hemisphere summertime, when NZ is on daylight saving time, for one hour each morning in NZ, from midnight to 1am, the day on Howland and Baker Islands is two days behind the day in NZ. eg if it is 0015 in Monday in NZ, it is 23:15 on Saturday (not Sunday) on Baker & Howland.
However, getting from NZ to Baker or Howland within the one hour that they are two days apart would require a suborbital rocket 'hop' . No other man-made transport is fast enough.